CONTINENTAL SHELF
RESEARCH
PERGAMON
Continental Shelf Research 20 (2000) 1433-1459
^
Characterization of intertidal flat hydrodynamics
P. Le Hira’*, W. Roberts'5, O. Cazaillef, M. Christie0,
P. Bassoulleta, C. Bachere
AI F REM ER, Centre de Brest, DEL/EC-TP, BP 70, 29280 Plouzané, France
hHR Wallingford, Howbery Park, Wallingford, Oxfordshire, 0X 10 8BA, UK
cSOGREAH Ingénierie, B.P. 172, 38042 Grenoble Cedex 9, France
dInstitute o f Marine Studies, University o f Plymouth, Plymouth, PL4 8AA, UK
eCREMA, B.P. 5, 17137 UHoumeau, France
Received 28 M ay 1999; received in revised form 27 January 2000; accepted 28 January 2000
Abstract
The paper reviews the different physical forcings that control tidal flat hydrodynamics. Tidal
propagation and cross-shore or long-shore currents, tidal asymmetry, wind-induced circula­
tion, wave propagation and drainage processes are successively considered. Some simple
methods are described for estimating cross-shore currents and wave-induced bottom shear
stresses, and the results obtained are compared to field measurements on three contrasted sites
in Europe. In particular the cross-shore current is shown uniform in the lower p art of the flat,
and decreasing towards the shore. Bottom friction-induced wave attenuation is simply for­
m ulated on gently sloping beds, leading to a maximum wave height that a flat can experience;
it is proportional to the water height according to the ratio between the slope and the wave
friction factor. The maximum related shear stress occurs at high water and is also proportional
to the water depth. Maximum tidal velocities are very similar in the three sites where bottom
sediment is muddy, suggesting a relationship between physical stresses and sediment character­
istics. The consequences of physical forcings on sediment transport are listed. The bottom shear
stress is suggested as the relevant parameter for comparing tidal and wave effects. In general, tide
induces onshore sediment transport, whereas waves and drainage favour offshore transport. The
processes leading to a possible tidal equilibrium profile are analysed: they involve the intrinsic
asymmetry that favours net deposition at high water, and an ebb dominance generated by the
resulting bottom profile convexity. Eroding waves are likely to upset such a balance; this equilib­
rium then reduces to a trend for the system. © 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Hydrodynamics; Tidal flats; Waves; Drainage; Tidal asymmetry; Energy dissipation; Bottom
stress; Mudflat; G eom orphology
* Corresponding author. Tel.: + 33-298-22-43-40; fax: + 33-298-22-45-94.
E-mail address: [email protected] (P. Le Hir).
0278-4343/00/$-see front m atter © 2000 Elesevier Science Ltd. All rights reserved.
PII: S 0 2 7 8 - 4 3 4 3 ( 0 0 ) 0 0 0 3 1 - 5
1434
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1. Introduction
Tidal flats are characterized by rapidly varying w ater depths, both in tim e and
space, w ith the additional air exposure. These strong w ater elevation fluctuations are
likely to induce specific hydrodynam ics, which in tu rn generate particular sedim ent
processes. H ydrodynam ics consist of b o th the flow and w ater depth tim e/space
evolution, w hich are generally strongly correlated (onshore current w hen w ater
elevation rises, offshore curren t w hen it falls). These forcings are responsible for
advection and dispersion, b u t they also generate b o tto m shear stresses, which are
relevant for sedim ent erosion an d deposition. T he b o tto m shear stress is the com bined
effect at the bed of non-linear in teractions betw een the m ean flow and the waves.
A review of physical processes an d their effect on sedim ent tra n sp o rt in intertidal
areas has been recently presented by Eism a an d R idderinkhof (in Eism a, 1997). The
m ain forcings th a t a coastal system can experience are (1) the tide, (2) the windinduced circulation, (3) the waves, (4) the density-driven circulation and (5) the
drainage process. D rainage processes are specific to intertidal areas w hen they are
exposed an d subjected to surficial w ater discharge.
Tide is naturally the m ost im p o rta n t process, as it determ ines the existence of the
intertidal flat. T idally induced currents can be split into a cross-shore com ponent,
w hich accounts for the filling an d the em ptying of the flat, and a long-shore com pon­
ent, w hich depends on the large-scale circulation aro u n d the flat.
Meteorological events are likely to generate large-scale variations of w ater surface
elevation, an d flow p attern s th a t m od u late an d som etim es counteract the tidal
currents.
Waves are either due to the p ro p ag atio n of offshore waves or generated by local
winds. Even in sheltered areas, waves are seldom negligible, as small waves can be
sufficient to resuspend soft cohesive sedim ents in very shallow areas, and often
contribute to m orphological stability in the long-term .
M any tidal flats are connected to estuarine systems where the salinity gradients
induce a density-driven circulation th a t frequently dom inates the tidal residual circula­
tion. A nother density effect occurs if there are strong tem perature variations a t the
m udflat surface due to solar rad iatio n (e.g. G uarin i et ah, 1997). These tem perature
variations m ay induce rapid changes in the local w ater tem perature (a 3°C change in
1 h has been m easured in a flat of the H u m b er estuary) and then therm al gradients
m ay appear over the flat. H ow ever, the density gradients have baroclinie effects th at
becom e im p o rta n t in deep areas only. T hus over a tidal flat, the density-driven
circulation can only develop aro u n d a connection w ith a neighbouring channel, and
then it is m ainly controlled by the large-scale circulation. F o r this reason, this forcing
will n o t be considered in the present paper.
The last m entioned forcing is drainage. This encom passes b o th the flow of surficial
freshw ater from upstream , an d the expulsion of porew ater from the upper sedim ents
as the w ater table low ers during falling tide. This latter process is favoured on m uddy
tidal flats where low perm eabilities prevent the porew ater to flow w ithin the bed.
Little is know n ab o u t the effects of drainage processes upon tidal flat evolution,
although observations indicate the im portance of drainage effects for the sedim ent
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1435
tran sp o rt (e.g. B assoullet et al., 2000). D rainage also appears an im p o rtan t factor for
long-term stability, w hen d rainage-m odulated seaw ards tran sp o rt com pensates for
a tidally induced residual accretion of the flat (Bessineton, pers. com.).
Assessing the co n trib u tio n of these five forcing processes to the b o tto m stress is
a difficult problem , n o t specific to tidal flats. H ow ever, for the intertidal zone, bed
irregularities are m ore likely, because of drainage, drying and swelling events in
m uddy areas, b io tu rb atio n by benthic anim als an d p rotection by plants. In addition,
the com p u tatio n of shear stresses is com plicated by extrem ely shallow w ater depths,
for w hich classical laws are questionable.
The purpose of this paper is to characterize the w ater m ovem ents (waves and
current) over in tertidal areas, considering in tu rn tidal pro p ag atio n and induced
currents, w ind-induced circulation, waves and drainage.
As show n by Friedrichs an d A ubrey (1996), the hydrodynam ics and m orphodynam ics are interrelated. This study focuses u p o n the dependence of the flows
or waves on the to pography, w hilst the in teractions betw een the hydrodynam ics and
sedim ent tran sp o rt, leading to possible equilibrium profiles, are analysed in detail by
R oberts et al. (2000).
2. Investigated sites
The w ork is m ainly based on observations an d co m putations on three sites which
have been investigated w ithin the IN T R M U D project, funded by the E uropean
C om m unity. By considering the consequences of hydrodynam ics on sedim ent
dynam ics an d m orphological equilibrium , the paper contributes to the IN T R M U D
classification of in tertidal flats (Dyer et ah, 2000). T he three investigated sites are
(Fig. 1):
( 1) the B rouage m udflat in the M aren n es-O léro n bay, on the w estern coast of France;
the in tertidal area is 4 km wide, w ith a m ean b o tto m slope of 1/800, w hich is
consistent w ith a tidal range of 5 m; the area is sheltered from A tlantic swells by
the Ile d ’O léron; the flat is entirely m uddy, w ith ridges and runnels on the flatter
section (slope « 1/1400) in the m iddle of the flat (Bassoullet et ah, 2000).
(2) the Skeffling m udflat 10 km inside the m o u th of H um ber estuary (England), where
it opens into the N o rth Sea. T idal range is 6 m, flat w idth is 4 km and the m ean
slope is 1/700, b u t in the order of 1/1300 in the m edium section of the flat. The
sedim ent is p redom inantly m uddy on the u pper part, b u t becom es mixed and even
sandy tow ards the low er levels. Intriguingly, this flat exhibits sim ilar shore norm al
ridges an d runnels in the m iddle of the profile (W hitehouse et ah, 2000) as the
B rouage m udflat.
(3) the N o rth e rn intertid al flat of the Seine estuary, near Le H avre (France), connected
to the English C hannel. T he area is m acro tid al (range = 7 m) and the tide is
characterized by strong asym m etries an d a large d u ratio n of high w ater slack
(about 3 h). T he m udflat slope (1/150) is larger th a n in the previous flats and the
waves are stronger, as the site rem ains open to the extended Baie de Seine.
P. Le Hir et al. / Continental S helf Research 20 (2000) 1433-1459
1436
North
Sea
S k efflin g
M udflat
4
f Spurn
.
Head
o
N
10 KM
Atlantic
Ocean
©
□
t
Humber
Estuary
0
Le Havre
) km
Northern Mudflat
Orry
Seine
Pont de
Norm andie
Là Roche/fe
-N46 00'
Ile
d'Oléron
Brouage
JVIudflal f.,j“
-N4S 50'
Marennes
5 km
W1 30'
i
t
W1 00'
Fig. 1. Location of the 3 investigated sites: (1) Skeffling mudflat in the H um ber estuary (LTK); (2) N orthern
m udflat in the Seine estuary (Fr.); (3) Brouage mudflat in the M arennes-O léron Bay (Fr.); D and B are the
m easurement stations.
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1437
Sedim ent is sandy in the low er p a rt an d m ixed or m uddy everyw here else on the
flat, b u t an n u al m onitoring m easured frequent changes in the sedim ent nature.
The following text gives an analysis of different hydrodynam ic features, m ostly
based on d a ta from these sites. These d ata can be records of flow velocity and pressure
(giving the tide an d waves) at fixed stations (e.g. Bassoullet et ah, 2000; Christie et ah,
1999), or results of depth-averaged modellings. The latter have been perform ed w ith
the SA M -2D H m odel (Brenon an d Te H ir, 1999) for the M arennes-O léron bay, and
the Télém ac-2D m odel for the H u m b er estuary (Roberts and W hitehouse, 2000) and
for the Seine estuary.
3. Tidal flow
3.1. Tide propagation
The wave length of the tide is m uch larger th a n the flat w idth so th at the w ater
elevation is nearly horizontal a t the scale of the flat. This feature can be w eakened by
two factors: first the tidal wave which p ropagates at the celerity (gh)0'5 can be retarded
in very shallow w aters (e.g. a t the tidal front), b u t usually only for a very short period.
T hen the surface elevation m ay present a significant slope, due to a supercritical
regime. Indeed, sm all bores are som etim es observed on tidal flats (e.g. in D ronkers,
1986), b u t these bores only affect the leading w ater’s edge. Secondly, b o tto m friction
can also slow dow n the tidal wave p ropagation, especially in shallow w aters and
across gentle slopes (e.g. Friedrichs an d M adsen, 1992).
These effects have been considered for the cross-shore pro p ag atio n of the tide. Even
w hen the tide is m ainly p ropagating along the shore, the phase lag betw een the shore
and the deep area off the flat rem ains controlled by the cross-shore propagation.
A one-dim ensional m odel th a t solves the depth-averaged shallow water equations has
been used. T he convex b o tto m profile is the result of m orphological com putations (see
R oberts et ah, 2000) an d is typical of tidally influenced m udflats (Friedrichs and
A ubrey, 1996; Tee an d M ehta, 1997). Fig. 2 shows the slight deviation of the surface
elevation from the horizontal, an d the currents associated to the tide propagation.
N oticeable surface slopes only occur behind the tidal front. W hen accelerations are
neglected, the resolution of w ater volum e conservation betw een two successive h o ri­
zontal w ater surfaces leads to depth-averaged currents very sim ilar to the ones
com puted by the shallow w ater m odel. This validates the hypothesis of nearly
instantaneo u s cross-shore tide p ropagation.
The flow field can be split into a cross-shore com ponent and a long-shore one, the
latte r being m ore related to large-scale processes an d generally stronger.
3.2. Cross-shore tidal currents
The m agnitude of the cross-shore curren t depends m ainly on the w idth of the
m udflat. It can exceed the long-shore current w hen the intertidal flat is particularly
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1438
|cross-shore depth-averaged velocity (m/s^
0.5
0 .3 -
0 .1 -
- 0.1
- 0.3
^A/ater Elevation (m)]
i.0n
5 .0 -
4 .0 -
3 .0 -
2 .0 -
0.0
0
1
2
3
4
5
6
7
8
9
10
Cross-Shore Distance (km)
Fig. 2. Cross-shore sim ulation of tide propagation and induced currents over an intertidal flat. D otted lines
are ebb/falling episodes. Thick curves are envelopes of m aximum flood or ebb velocities over the tide.
wide (e.g. the B rouage m udflat), a n d /o r w hen the alongshore currents are reduced by
the presence of a n atu ral or m an-m ade obstruction, such as a headland, spit or
b reakw ater (e.g. the Skeffling flat, Fig. 3).
In the case of uniform b o tto m slope ß , it can be show n (Appendix A; Friedrichs and
Aubrey, 1996) th a t at any location of the flat below m ean w ater level, the m axim um
cross-shore curren t is approxim ately
ßT,
d)
where R and T tide are, respectively, the tidal range and period. In practice, the slope of
m any estuarine m udflats aro u n d the m ean w ater level is flatter th an the average slope
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1439
> 1 m/s
1km
Fig. 3. Com puted current field on the Skefiling flat (Humber estuary), 2h30’ before high water. Bathymetry
contours shown are LWS and MWL. (After Roberts and W hitehouse, 2000).
Table 1
Site
Tidal range (m)
Bottom slope
Fm ax
Brouage
flatter section
Skeffling
flatter section
Le Havre
5
1/800
1/1400
1/700
1/1300
1/150
0.28
0.49
0.29
0.55
0.07
6
7
(m S b
from high-to-low w ater, leading to a greater peak current th a n th a t indicated by
Eq. (1). F o r the three investigated sites, Eq. (1) gives the result of T able 1.
O n the u pper flat (above m ean sea level), Friedrichs and A ubrey (1996) show th a t
the m axim um current occurs at the tidal front, and the intensity is given by
Mmax=^
J
1 - ( ^ Xf) ’
(2)
where x f is the location of the tidal front (set equal to 0 at mid-tide).
F o r a linearly sloping flat, above the m ean sea level, m axim um cross-shore currents
decay when ap proaching the shore.
These schem atic results, which have been extended for m ore com plicated geom et­
ries by using hypsom etry d a ta (Pethick, 1980; Friedrichs and Aubrey, 1996), can be
com pared to field observations or co m putations from depth-averaged m odels of the
investigated flats. Fig. 4 gives representative exam ples of w ater level and current,
recorded at statio n B on the B rouage flat (Fig. 4(a)) or com puted on several locations
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1440
(a)
(b)
Water Elevation (m)
Water Elevation (m)
6.0
4.0 -
4.0
2.0
2.0
-
0.0
0.0
velocity 32 cm above bottom (m/s)
Depth-averaged velocity (m/s)
0.5
0 .4 -
0.2
0.0
T
0
2
4
6
8
TIME (hours)
10
12
0.0
25
27
29
31
33
35
37
TIME (hours)
Fig. 4. Tide elevation and current velocities in the Brouage m udflat (M arennes-Oléron Bay, Fr):
(a) m easurements at station B, Nov. 1994; (b) com putations at 4 locations along a cross-section of the flat.
of the sam e flat (Fig. 4(b)). As stated before, the flow is m ainly cross-shore at this site.
A ccording to the m easurem ents as well as the m odel, m axim um flood velocity occurs
ju st after covering, approxim ately a t m id-tide; after a long high w ater slack, the ebb
progressively increases until a m axim um speed ju st before uncovering. This co rro b o r­
ates the fact th a t m axim um velocities are reached at m id-tide, or near the tidal front
in the upper flat. The m axim um velocity is ab o u t 0.45 m s - 1 , in agreem ent with the
order of m agnitude of a depth-averaged velocity deduced from Eq. (1). In addition,
the m odel produces a relatively uniform m axim um velocity in the lower p a rt of the
flat (up to mid-level) and a low er velocity in the upper flat, as indicated by relations
(1) and (2).
Fig. 5 presents the com puted m axim um velocity (over a m ean spring tide) at several
locations for the three investigated sites versus the “reduced” m axim um w ater depth.
The latter is defined as the m axim um w ater height over the tide divided by the tidal
range and represents a norm alized distance to the shore when the b o tto m slope is
uniform . In co n trast w ith the H um b er and B rouage results, on the Seine site velocities
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1.2
T
♦ Humber
1
Flood
■ Seine
0.8
0.6 If)
1441
Brouage
0.4
E
0.2
o
o
reduced maximum water depth
------------------------
0 -
2 -0.2 -
*1 0.5
9
-0 4 -
0.6
-
0.8
Ebb
-
Fig. 5. D istribution of com puted maximum velocities at different locations of the three investigated sites
(mean spring tide). The “reduced maximum water depth” is the local maximum w ater height over the tide,
divided by the tidal range.
increase w ith distance from the shore, even in the lower flat. In addition these
velocities are considerably higher th a n the theorical cross-shore com ponent deduced
from Eq. (1). In fact, the flow is m ainly longshore on this flat, and thus controlled by
the large scale tidal p ropagation. H ow ever, in the upper part of the three sites, the
to tal velocities are rem arkably sim ilar, w hich p ro b ab ly explains the analogy betw een
the corresponding sedim ents. A ctually, the low er p art of the Seine flat is nearly sandy,
where the tidal flow is larger. Lastly, the three sites exhibit a sim ilar tidal asym m etry in
favour of flood, which will be discussed below.
3.3. Long-shore tidal currents
C urrents parallel to the shore are dependent on large scale topography. Typically,
the volum e of w ater w hich m ust pass th ro u g h an estuary cross-section during a tidal
cycle is determ ined by the tidal range an d the shape of the estuary landw ard of th at
section. The distribution of flow across the section can be approxim ated by relating
the current speed to the square ro o t of the w ater depth. This arises from balancing the
bed friction distributed over the w ater colum n (t//;) an d the longitudinal w ater surface
slope in the depth-averaged m om entum equation, assum ing this slope is uniform
across the section, which is reasonable w hen the flat is not too wide. M ore precisely,
the bed friction is generally form ulated as a q u ad ratic function of the depth-averaged
velocity, m o dulated by a function of the w ater height th at represents the integration of
velocity profile [for instance 1/Ln[h/ez0 ) for a logarithm ic profile]: as a result, the
dependence of current speed on the w ater height across the flat can be slightly larger
th an the square root.
The N o rth e rn flat of the Seine estuary is representative of relatively narrow
intertidal flats connected to a m acrotidal estuary where currents are strong.
Fig. 6 shows th a t sim ultaneous currents com puted on different stations of the flat are
actually correlated to the pow er 0.5 + 1/6 of the local w ater depth, which is in
P. Le Hir et al. / Continental S helf Research 20 (2000) 1433-1459
1442
(C)
1 T
♦ flood
0.8
0.6
E
o 0 .4
m
E
5c
<D
t 0.2
>s
Ö
o
o
CD
0
0
>
<.
£Q.
2
3
{Water H eight)0667
0
Û
1
34
36
38
40
42
44
46
TIME (hours)
Fig. 6. Com puted tidal characteristics along a cross-section of the N orthern flat in the Seine estuary:
(a) water elevation, with noticeable tide distortion; (b)depth-averaged velocity, with strong flood dom i­
nance; (c) correlation between velocity and w ater height, at typical instants of the tide.
agreem ent w ith the Strieker form ulation of the b o tto m friction (t ~ it2//i1/3) used in
the model.
This kind of relationship can be m odified to some extent by lateral diffusion, which
tends to partially even out current speeds across the estuary, and by w ind-induced
currents. Also, at the tim e of peak current in the m ain channel, w ater depth on the
m udflat is often very small, so th a t the low er in tertidal flat is m ore strongly influenced
by long-shore currents th a n the upper flat.
3.4. Tidal asymmetry
W e can distinguish three kinds of tidal asym m etry which are likely to induce a net,
tidally averaged sedim ent tra n sp o rt over a tidal flat:
(i) an intrinsic asym m etry due to the air exposure,
(ii) large-scale asym m etries due to tide p ro p ag atio n in the surrounding basin,
(iii) local asym m etries generated by the to p o g rap h y of the flat.
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1443
Types (ii) and (iii) induce differences betw een the average m agnitude of flood and ebb
currents, w hereas all kinds of asym m etries generate differences in the d u ratio n of the
slack w ater period, type (i) being the m ore relevant in the case of intertidal flats. As
pointed out by Eism a an d R idderinkhof (ibid) am ong others, the difference between
flood and ebb m agnitudes is m ore im p o rta n t for bedload tran sp o rt, whereas the
difference in slack d u ratio n s is fundam ental for suspended m atter transport.
The intrinsic asym m etry is merely the fact th a t the upper p a rt of an intertidal flat
experiences only one slack at high w ater. As a consequence, particles com ing onshore
during the flood begin to settle onto the upper flat w hen the b o tto m stress is below its
critical value for deposition. D eposition processes are favoured by the decreasing
velocities near the shore. This allows tim e for the partial consolidation of newly
deposited cohesive m aterial. T he subsequent resuspension of the deposited m aterial
requires the ebb flow speed to exceed some m inim um , and often this m inim um is
h ardly reached in the u pper flat except if the ebb current is locally dom inant. These
processes, know n as settling and scour lag effects, have been described by P o stm a
(1961).
The large-scale asym m etry is caused by the non-linear distortion of the tide during
p ro p ag atio n (advective an d friction term s in the m om entum equation), and is depen­
d ent on the geom etry of the coastal area (e.g. Uncles, 1981). W hen the flat is connected
to an estuary where the tide p ro p ag atio n is slowed dow n, the asym m etry is generally
of flood d o m in an t type. The tidal flow over the N o rth e rn flat in the Seine estuary (Fig.
6) exemplifies this kind of asym m etry. It should be noted th a t these large-scale
asym m etries m ay vary, depending on the h arm onic com position w ithin for instance
the fortnightly tidal cycle. This is illustrated in Fig. 7 which shows an inversion of the
flow asym m etry on the B rouage m udflat, w ith a flood dom inance at the beginning of
the selected period and an ebb dom inance a t the end, for sim ilar tidal ranges. A m ong
tidal asym m etries, residual eddies due to headlands or islands have also to be
m entioned, as they are likely to induce specific sedim ent tran sp o rt or deposition (e.g.
the P o rtish ead m udflat, in the Severn estuary, W hitehouse and M itchener, 1998).
The b o tto m to p o g rap h y of the flat can itself induce or m odify the tidal asym m etry.
Several au th o rs pointed out th a t tidal flats w ith a “convex” profile enhance the
m axim um ebb curent (e.g. D ronkers, 1986). A ctually the cross-shore com putations
plotted in Fig. 2 show an ebb dom inance th ro u g h o u t the profile, except near the
shore end.
T o sum m arize, it can be said th a t large-scale to p o g rap h y induces asym m etries of
the tide elevation and long-shore flow com ponent, generally in the way of flood
dom inance, w hereas local to p o g rap h y induces a cross-shore asym m etry.
3.5. Bottom shear stresses and energy dissipation
As stated in the intro d u ctio n , b o tto m stress is the relevant param eter for assessing
the deposition an d erosion of sedim ents. It can be expressed as a quad ratic function of
the m ean flow (or the flow at a given height), providing a drag coefficient
(or equivalently a roughness length if a logarithm ic velocity profile is assumed). This
has been extensively analysed by C ollins et al. (1998) w ho pointed out the strong
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1444
W ater Height (m)
4.0 —i
2 .0 -
0.0
velocity (m/s)
0.6
0 .4 -
0 .2 -
0.0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
TIME (days)
Fig. 7. M easured tide elevation and current velocity (32 cm above bottom ) at station B in the Brouage
m udflat over a fortnightly tidal cycle. Flood dom inance appears from the beginning until the 4th day,
whereas ebb is dom inant between the days 6 and 12.
variability of these param eters. A ctually the determ ination of drag coefficients on
intertidal areas is com plicated by the presence of bedform s, ridge and runnel n et­
w orks, channels, benthic fauna and plants. All these param eters can induce flow
p ertu rbatio n s th a t modify the velocity profile and prevent representative bed shear
stresses being calculated from local m easurem ents. W e thus prefer using the m odel
results for characterizing the current-induced shear stresses at the scale of the flats.
F o r instance, Fig. 8 gives the distrib u tio n of the m axim um b o tto m stress (over
a spring tide) on the Skeffling m udflat. This shear stress was com puted using a finite
elem ent depth-averaged m odel (TELEM A C -2D ), assum ing a uniform N ikuradse
roughness of 0.01 m. It can be seen th a t this tidal-induced peak shear stress is n o t
uniform over the flat: it presents a strong decay tow ards the shore in the upper flat
while it is m ore uniform in the low er half. These results are quite in agreem ent w ith the
cross-shore velocity gradients predicted from Eqs. (1) and (2). As for the long-shore
com ponents, it should be rem inded th a t they can be deduced from the hypothesis of
uniform longitudinal surface slope across the flat: then from the depth-averaged
m om entum equatio n in steady regime, it is concluded th a t long-shore b o tto m stress
com ponents are p ro p o rtio n al to the w ater depth and then also they decay shorew ards.
A nother com m only used param eter is the energy dissipation rate, assum ed to be
uniform w hen the flat is in equilibrium (e.g. Lee and M ehta, 1997). C om puted as the
w ork of the b o tto m stress, approxim ately p ro p o rtio n al to the cube of the flow speed, it
only enhances the co ntrasts presented by the b o tto m stresses. Finally, the d istribu­
tions of shear stress and energy dissipation rate b o th reflect the velocity gradients
and variations. The m ain advantage of these param eters over a simple velocity
P. Le Hir et al. / Continental S helf Research 20 (2000) 1433-1459
1445
Peak bottom
shear stress
(N/m 2)
■ s'°
™ 3.0
i “
I 0.5
‘
0.3
0.2
0.1
0.0
I km
Fig. 8. Com puted distribution of tidally induced shear stress over the Skeffling mudflat in the Flumber
estuary (maximum over a spring tide), (bottom roughness: N ikuradse coef. = 0.01 m).
distributio n is to enable a com parison w ith wave effects, and to be m ore relevant for
the sedim ent behaviour.
4. W ind-induced circulation
M eteorological events act on w ater systems th ro u g h atm ospheric pressure gradi­
ents and wind. P ressure gradients can induce large-scale variations of w ater surface
elevations. The tim e-scale of this process is generally too large for the related transient
currents to be im p o rtan t, but the sto rm surge can significantly change the w ater depth
on a flat (a few tens of cm, or even m ore th a n 1 m) an d consequently change the air
exposure as well as the local tidal flows. M o re frequently, winds are likely to generate
a specific circulation by entrainm ent of surficial w aters and generation of an ad d i­
tional surface slope, which in tu rn induces retu rn flows in deeper waters.
In order to illustrate these w ind effects, sim ulations of storm y winds over the
B rouage m udflat have been perform ed, by using the depth-averaged m odel of the
M arennes-O léron Bay. A steady w ind of 20 m s “ 1 was blow ing eastw ard for 5 h either
during a rising tide or a falling tide. Changes are noticeable on the upper flat (Fig. 9).
A lthough the elevation is only enhanced by ab o u t 0.1 m, the current speed is roughly
doubled an d asym m etry can reverse. N evertheless at this location the b o tto m stresses
rem ain small enough not to induce resuspension, so th at the m ain consequence of the
w ind entrainm ent will be a m odification in residual sedim ent tran sp o rt and depos­
ition.
N aturally, local winds generate waves th at are likely to resuspend sedim ent, so th at
the correlation betw een the w ind an d wave episodes should be accounted for.
1446
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
6.5n
Water Elevation (m)
6 .0 -
5.5-
5.0
4.5
0.5n
Depth-averaged velocity (m/s)
0.40.30 .2 -
0.1-
0.0
25
27
29
31
33
35
37
TIME (hours)
Fig. 9. Simulation of wind effects on hydrodynamics on the upper Brouage mudflat: (a) spring tide, without
wind; (b) same tide, 2 0 m s “ 1 western wind blowing 5 h during the rising tide; (c) same tide, 2 0 m s “ 1
western wind blowing 5 h during the falling tide.
5. W aves
F rom one tidal flat to another, the wave regim e can be very different: either the flat
is exposed to offshore swells, or it is only subm itted to short waves generated by local
winds. In the first case, the flat is generally sandy and is nothing else th a n a beach.
W hen nevertheless the area is m uddy, because of a large in put of sedim ent nearby,
m ud liquefaction is likely to occur and the waves experience strong dam ping (e.g.
Wells and K em p, 1986; Li and M ehta, 1997). In the second case, small waves can have
an im p o rta n t influence on sedim ent tran sp o rt, because, in shallow waters, the stress
they cause on the bed is com parable w ith the stress associated with a quite substantial
tidal current.
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1447
' ++ X
0
1
2
3
4
Water depth (m)
Fig. 10. C orrelation between wave height and water depth m easurements in the Brouage m udflat (France).
Crosses: at station D, 17/26 Nov. 1997; dots: at station B, O ct-N ov. 1994; “rms wave height” is estimated as
21-5 times the standard deviation of near bed pressure fluctuations. Dashed line: H rms = 0.15 h.
5.1. Wave height evolution on sheltered tidal flats
The size of locally generated waves is dependent on the fetch length, the wind speed
and the length of tim e for w hich the w ind blows. F o r a fetch length of 5 km for
example, the length of tim e for the waves to reach their fully developed state is around
30-40 mins, depending on the wind speed. In this respect, tim e is n o t usually a lim iting
factor, the wind speed, direction and fetch length are m ore im portant. F o r a given
wind speed, wave height increases approxim ately as the square ro o t of the fetch
length. F o r a fixed fetch length, the wave height increases approxim ately linearly with
wind speed. Then, when pro p ag atin g over the flat, waves are likely to increase in
height due to shoaling and fetch extension, and to decrease, due either to b o tto m
friction or to visco-elastic entrainm ent of the bed.
Fig. 10 presents an overview of “ro o t m ean square” wave heights m easured on two
locations of the B rouage m udflat, during 20 days at station B on O ctober 1994, and
during 9 days at statio n D in N ovem ber 1997. The ro o t m ean square wave height
H rms is com puted as 23/2 times the stan d ard deviation of the pressure fluctuations near
the bottom , neglecting the wave pressure decay over the w ater colum n. It is striking
th a t in b o th cases the p lo t exhibits a linear relationship betw een the m axim um wave
height and the local w ater depth, w ith the sam e slope at tw o different locations of the
flat. W hatever the depth, the significant wave height does n o t exceed 15% of this
depth. This can be com pared to the ratio of 0.19 betw een H rms and the w ater depth
observed on the Surinam coast by Wells and K em p (1986). They observed th a t the
waves never broke because their height rem ained below the usual breaking lim it
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1448
(H /h ~ 0.8), an d they attrib u ted the severe wave atten u atio n to dissipation into the
b o tto m by the generation of a m udw ave. F lu id m uds are frequently observed near the
sea bed along the Surinam , an d m ay correspond to the case of exposed coasts
previously m entioned. Flowever, in the case of the B rouage m udflat, the liquefaction
of the b o tto m does n o t seem to occur, at least under small or m edium waves.
The reason for this “sa tu ra tio n ” of the local wave height on flats can be looked for
in b o tto m friction. Follow ing the linear theory an d applying the conservation of
energy for a m on o ch ro m atic wave p ro p ag atin g on a uniform slope w ith q u adratic
b o tto m friction, a relation can be found (see A ppendix B) betw een the wave height
H and the distance from the shore x according to the differential equation
—
dx
( H 2 x 112) = — -
37i ß
(3)
where / w is the wave friction factor.
This eq u atio n can be conveniently non-dim ensionalized according to
L - ( H 02 X 1l2) = ^ - f- j H l X - ^ 2,
(4)
where H 0 = H /h 0 an d X = x ß /h 0, w ith h0 the w ater depth at the offshore lim it of
investigation. h0 should n o t be too large (typically 5 m) because of the shallow w ater
approxim ation.
Eq. (4) has been num erically integrated, an d the solution is dependent on the
relevant non-dim ensional p aram eter f w/ß (Fig. 11). It can be seen th a t the trend of the
wave atten u atio n is very different according to the value of this param eter. F o r low
values off w/ß (small friction or large b o tto m slope), dissipation is negligible and wave
height is m aintained or even increased (shoaling) until breaking, defined here as
a m axim um H /h classically equal to 0.8. O n the other hand, for large values o ff w/ß
(high friction or gentle slope), dissipation becom es d o m in an t and the wave height
tends to a co n stan t p ro p o rtio n of the w ater depth, w hatever be the incident wave
height. T hen this p ro p o rtio n gives the m axim um wave height such a tidal flat can
experience at a given w ater depth. A ssum ing this p ro p o rtio n (H/h)iim exists, it can be
com puted from Eq. (3) by replacing H by h(H/h)iim or ßx(H/h)iim, w hich after
derivation leads to
.t« / ) Um
-
¥4
Jr w
W e assum e th a t Eqs. (3) and (5) can be used w ith H rms instead of H, although the
constant resulting from the integration of the dissipation rate could change according
to the wave height distribution. T hen the application of relation (5) for the case of the
B rouage m udflat (HrmJ h ) iim = 0.15 im plies th a t the wave friction factor should be
approxim ately 0.05.
M easurem ents are also available in the upper section of the N o rth e rn m udflat in the
Seine estuary. W aves and w ater depths have been recorded for 2 days during a storm
(Fig. 12; Silva Jacinto et ah, 1998). These m easurem ents also show a linear relationship
P. Le Hir et al. ( Continental Shelf Research 20 (2000) 1433-1459
1449
(a)
0.9 -
0.8
-
0.7 -
S
— Hsea /
■■■■ Hsea /
— Hsea /
- - Hsea /
--H s e a /
— h/hO
hO =
hO =
hO =
hO =
hO =
.1
.2
.3
.4
.5
0.5 0.4 0.3
0.2
-
0
0.2
0.4
0.6
0.8
1
distance offshore * slope / hO
(b)
0.8
-
o
_-i—
£=< 0.7 -
— Hsea /
---- Hsea /
— Hsea /
-■ Hsea /
--H s e a /
— h/hO
hO =
hO =
hO =
hO =
hO =
.1
.2
.3
.4
.5
£03 0.5 5
o3
0.4 —
o
sz
0.3
CD
>
03
£
0
0.2
0.4
0.6
0.8
1
distance offshore * slope / hO
Fig. 11. Com puted wave propagation over a linearly sloping bed (Eq. (4)) for different incident wave
heights. The line “h/ho” represents the non-dimensionalized water depth, (a) f w/ß = 50 (wave friction
factor = 0.05, slope = 1/1000) (b) f w/ß = 10 (wave friction factor = 0.05, slope = 5/1000).
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1450
0.8
0.7
<o 0.6
0.5
o>
0.4
0.3
0.2
0.5
1
1.
2.5
W a te r d e p th im)
Fig. 12. C orrelation between root m ean square wave height and water depth, measured in the N orthern
m udflat of the Seine estuary (France), during a storm (12-14 feb.1997). (after Silva Jacinto et al., 1998).
betw een H rms and but the ratio is 0.28, which is in agreem ent w ith the larger bo tto m
slope of this flat. A ccording to (5), the m ean wave friction factor should be 0.14, which
is close to the upp er lim it of the usual range (from 0.005 to 0.4, Soulsby et ah, 1993).
If the seabed consists of soft m ud, then a significant p ro p o rtio n of the wave energy
m ay be absorbed as it progresses across the bed. Lee an d M ehta ( 1997) have addressed
this case and discussed m echanism s of sedim ent tra n sp o rt by waves over a soft m uddy
bed, in the case of small tidal range.
5.2. Wave-induced bed shear stress
C om bined w ith the shallow w ater ap p ro x im atio n of the orbital velocity, the
expression of the w ave-induced shear stress (Eq. (3), A ppendix B) becomes
_
T
P 0 Íw H 2
8
h'
A ssum ing a linear relationship betw een the m axim um wave height and local depth,
the bed shear stress reads
pgfv [ m m lH iA H /h h J t) ] 2
,T
'
161
where
is the incident wave height. Shorew ard of the zone where the m axim um
wave height to depth ratio is achieved, the shear stress decreases linearly w ith depth,
as the wave m oves tow ards the shoreline in the depth-lim ited zone. Seaw ard of the
sam e tran sitio n zone, the stress variation is inversely pro p o rtio n al to the depth. This is
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1451
Water Elevation (m)
Water Elevation (m)
i.0—i
¡■On
transition
level
<
4 - transition level -►
2.0
2.0
0.0
0.0
Bottom Shear Stress (N/m2)
5.0
Bottom Shear Stress (N/m2)
5.0
4.0
4 .0 -
3 .0 -
3 .0 -
2 .0 -
2 .0 -
1 .0
0.0
-
0.0
25
27
29
31
TIME (hours)
33
35
37
25
27
29
31
33
35
37
TIME (hours)
Fig. 13. W ave-induced bottom shear stress computed on different locations of the Brouage mudflat
(M arennes-Oléron Bay, France), according to Eq. (6). Time evolution over a spring tide, (a) (left): incident
wave height = 0.5 m; (b) (right): incident wave height = 0.2 m.
The “transition level” separates the upper fraction of the flat where the “saturation” wave height is
reached (whatever be the incident wave) and the lower part where the m aximum stress is uniform and
depends on the incident wave.
show n in Fig. 13, which plots Eq. (6) at different locations of the B rouage m udflat
during a spring tide, taking (H/h)Um = 0 .1 5 . It should be noted that:
• the tran sitio n betw een the tw o trends is dependent on the value of the incident wave
height H o, for a given b o tto m slope;
• the upper flat experiences the highest stresses a t high tide, w hatever be the incident
waves;
• the m iddle flat experiences the highest stresses a little after covering or before
uncovering, when the w ater depth is
; for a given incident wave, this
m axim um stress is uniform;
• for a given b o tto m slope, the m axim um stress th a t any location can experience is
reached at high w ater w hen
= (H/h)UmhHW, then it linearly increases w ith the
w ater depth at high w ater hHW;
1452
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
• an increase of the b o tto m slope will delay or suppress the linear relationship
betw een the stress an d the w ater depth.
T he order of m agnitude of w ind-induced stresses is at least the same as the one related
to tidal flows. A m ajo r difference comes from th a t the form er m ay be strong at high
w ater, even in the upper flat, w hen the latter vanish. The consequence on sedim ent
tra n sp o rt will be analysed in the discussion.
6. D rainage
A lthough drainage processes are com m only observed features on exposed tidal
flats, very few studies have investigated this type of process. D rainage can be either
due to creeks flowing from the shore at the surface of the flat (e.g. W hitehouse et ah,
2000), or due to dew atering w ithin the air-exposed sedim ent, w hen the w ater table is
falling w ith some delay after the w ater surface (e.g. A nderson and Howell, 1984).
The w ater table gradient is supposed to generate pore-w ater flow inside the
sedim ent, b u t in m uddy environm ents, the low perm eability forces the pore w ater to
escape th ro u g h the interface. W hen the b o tto m slope is rath er steep (in the order of
1 % or more), the w ater flows rapidly an d evenly on the m ud. W hen the slope is gentle,
surficial flow is slow an d a criss-crossed p attern of runnels which intersect each other
is formed. These runnels grow an d becom e organized progressively while running
seaw ard, then form ing a netw ork of parallel ridges and runnels, cross-shore orien­
tated. These pattern s can be observed in the B rouage m udflat as well as the Skeffling
m udflat (e.g. W hitehouse et ah, 2000). T he form ation of the runnels by dew ateringinduced surficial w ater flows is n o t clearly established, b u t the co n trib u tio n of these
longitudinal bedform s to drainage is u n doubted, despite the fact th a t they retain w ater
in the runnels at the end of the exposure period. As w hen the b o tto m slope is gentle
cross-shore currents are dom inant, the runnels are o rientated along the tidal currents
(and m ost often along the d o m in an t waves, due to wave refraction). It seems th a t the
tidal current enhances the organisation of runnels in a parallel netw ork, contrasting
w ith the m eandering system w hich first occurs, w ithout surprise over a flat
bathym etry. T he fact th a t ridge an d runnel p attern s extend in the section of the flat
where the slope is m inim um , i.e. w here currents an d b o tto m shear stresses are stronger
(Eq. (1)), is notew orthy.
T o the a u th o rs’ knowledge, there has been no com prehensive investigation on the
quantification of drainage-induced w ater discharge. If dew atering of the upper section
of the flat is the only source, the discharge can be evaluated from the decrease of
pore-w ater content of the m ud, or even from the decline in sedim ent elevation.
A ssum ing a m ean reduction of Sh = 1 cm for the sedim ent level during half of the
m ean air-exposure time, i.e. a b o u t St = IO4 s, the local sink of pore w ater correspond­
ing to the sedim ent variation is Qs = Sh = 0.01 m 3 m ~ 2. After a cross-shore distance
d = 2000 m from the shore, the induced discharge can be evaluated as
Qd = Qs-d/St = 0.002 m 2 s “ 1. This m eans th a t to evacuate the pore w ater as surficial
flow, the latter should be in the order of 0.002 m 2 s _1 per unit w idth. If this flow is
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1453
concentrated in runnels, w ith an active cross-section A = 0.2 m x 0.1 m (where the
w ater flows), every 2 m (the distance X betw een runnels), the current needed in runnels
for discharging pore w ater is U d = XQd/ A = 0.2 m s - 1 . This is actually the order of
m agnitude of velocities th a t are observed in the runnels. T hey are n o t negligible, and
likely to tra n sp o rt suspensions and to prevent consolidation in the runnels.
N atu rally runnels are n o t the only w ay of drainage. C hannels are quite com m on
features an d contribute to drainage. They often coexist w ith runnels, and b o th systems
m ay exhibit a slight angle betw een them , runnels being m ainly parallel to the tidal
currents an d channels converging to the bathym etric gradient (Bassoullet et ah, 2000).
7. D iscussion
M ost of the previous investigations result from sim plifications of real cases, where
generally all processes are com bined. M athem atical m odels are the only tool which
enables us to account for all these processes w ith a realistic topography. They are
presently usable, at least for hydrodynam ics, an d som e applications have been show n
here. B ut our purpose in this paper is to point out the m ain features th a t we can simply
deduce from the know n characteristics of a given intertidal flat, w ithout running
a m odel which will take a long tim e to im plem ent. This justifies the approxim ations
for the tide pro p ag atio n, an d the choice of the sim plest m odel for the waves.
How ever, n o t all the processes have been considered. F o r instance one can w onder
how im p o rta n t is the rain, either as an additio n al source of w ater for drainage, or as
a m echanism for roughness changes (e.g. A nderson, 1981). Similarly, fauna-induced
b io tu rb a tio n can change the roughness. The ab undance of algae and plants is
probably m ore essential, as they can slow dow n the w ater flows over the flat, and then
favour sedim ent deposition (trapping). These “n a tu ra l” processes and their conse­
quences have been reviewed by A nderson (1981). O ne im p o rtan t feature is the
seasonal variability th a t these biotic com ponents can induce, and its consequence on
sedim ent tra n sp o rt is becom ing a topical subject th a t is beyond the topic of this paper.
M an m ay also m odify the tidal flat hydrodynam ics. F o r instance, shellfish farm ing
installations, frequently set on the lower flats, are likely to change the current
orientatio n an d to dam pen waves. This appears to be the case in the M arennesO léron Bay, an d p artly explains the wave atten u atio n observed on the B rouage
m udflat. Tittle is know n ab o u t the quantification of these processes. B oat waves
constitute an o th er forcing w hich is n o t very different from w ind-induced waves, except
th a t it can occur in sheltered areas close to channels, as for instance in upper estuaries.
7.1. Tidal fo rcin g and sediment response
Back to the tidal forcing, we can consider the consequences on sedim ent processes
of the different kinds of asym m etry th a t have been pointed out:
• the intrinsic asym m etry, which results from a single tidal slack between flood and
ebb experienced by the u pper flat, is responsible for a net sedim ent tran sp o rt
1454
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
tow ards the shore, providing m aterial is available (from the low er flat or from
offshore); scour-lag effects (Postm a, 1961) induce a net deposition on the upper flat,
which favours the convexity of the b o tto m profile;
• the shape of the cross-shore profile can generate tidal current asym m etries; in
particular, a tide-related convex profile enhances the m axim um ebb current; the ebb
current enhancem ent favours the seaw ards export of sedim ent, in co ntrast w ith the
shorew ards tra n sp o rt induced by the intrinsic asym m etry; this can lead to some
sedim entological equilibrium , as indicated by m orphological m athem atical m odels
under developm ent (e.g. R oberts et ah, 2000); for example, the convex bo tto m
profile show n in Fig. 2, for which tidal currents exhibit an ebb dom inance, results
from the evolution of an initially linear profile over a period of 17 years, sim ulated
by a cross-shore sedim ent tra n sp o rt model;
• large scale asym m etries are often related to the tide pro p ag atio n in an estuary to
which the flat is connected, and generally are flood dom inant; at equilibrium , one
can w onder w hether this flood dom inance should n o t be com pensated by an
increase of cross-shore convexity th a t w ould reinforce the ebb.
It can be n o ted th a t the tide-generated convexity com pensates for the decay of
m axim um tidal shear stresses in the upper flat, an d finally tends to m ake it uniform.
T his feature is p ostulated by Friedrichs an d A ubrey (1996) am ong others to get
sedim entological equilibrium . T hen studying the case of non-rectilinear shore, they
concluded th a t the profile convexity was enhanced w ithin an em bayed shoreline, and
reduced w hen the latter is lobate.
Conversely, sedim ent features can give som e insight into the hydrodynam ics c h ar­
acterization of a tidal flat. A com parison betw een the steep N o rth e rn flat in the Seine
E stuary and the B rouage or Skeffling m udflats, whose slopes are gentle, shows th at
m axim um velocities are sim ilar (around 0 . 4 m s _1, Fig. 5), except in the low er p a rt of
the Seine flat w here they are larger. These larger velocities ( l m s -1 or so) correspond
to the sandy area of the flat, w hereas the rest is m ainly m uddy, as the other
investigated flats. This confirm s a relationship betw een the natu re of sedim ent and the
m axim um tidal velocity, as p ostulated by m any authors. As far as cohesive sedim ent is
concerned, this relationship should be based on the ratio betw een the induced
m axim um shear stress and the critical shear stress for erosion. Then, assum ing (i) fine
sedim ent is available from offshore or from a river in p u t and (ii) the flat is n o t far from
equilibrium , a m axim um velocity can be estim ated from the sedim ent nature. By using
Eq. (1) for cross-shore currents (w ~ tidal range/slope) and the dependence of long­
shore current on the square ro o t of local depth, the slope of the flat can be iteratively
deduced from the tidal range an d the long-shore current off the flat. Such an
adaptability of the slope seems to be confirmed by recent m orphological studies (e.g.
Friedrichs and Aubrey, 1996; Lee and M ehta, 1997; R oberts et a l, 2000). In other words,
the com bined inform ation of topography, long-shore current at the boundary, tidal
range, tidal velocities and sediment nature seems redundant. This result, deduced
w ithout accounting for wave effects, can be used to describe a lack of balance —
possibly due to wave erosion — and thus predict a m orphological trend. Also this kind
of redundance should be kept in m ind when aiming at classification of intertidal flats.
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1455
7.2. Wave effects
W ave-induced stresses are large on tidal flats, so th a t erosion is likely to occur.
Small incident waves have m axim um effect w hen the local depth equals the “sa tu ra ­
tio n ” wave height, th a t is a t the beginning of local flood or at the end of ebb: then
sedim ent resuspended on flood is being tran sp o rted onto the upper flat where it can
deposit during slack (high) w ater, leading to a possible net accretion. O n the other
hand, high incident waves — which are often enhanced at high w ater w hen the fetch is
extended — have m axim um effect a t high water: in this case resuspended m aterial m ay
be tran sp o rted offshore by ebb currents an d a net erosion will result. These features
can change drastically depending on the direction of w inds th a t occur sim ultaneously.
F o r instance at the Skeffling m udflat, w ind velocities over th a n 5 m s -1 have been
proved to reverse the shorew ards tra n sp o rt of m ud (Christie et ah, 1999). N evertheless,
effects of strong waves am o u n t to global erosion an d then generate concave bo tto m
profiles, a shape th a t is consistent w ith uniform w ave-induced b o tto m stress (e.g.
Friedrichs an d A ubrey, 1996).
In m ost areas, b o th tidal flow an d wave forcings coexist. The question of their
relative im portance is often addressed. It can be assessed by b o tto m stresses co m p u ta­
tion, based on full tw o-dim ensional m athem atical m odelling, w hich is tim e-consum ­
ing especially for waves, or estim ated from the simple Eqs. (1) and (6), provided
continuous m easurem ents of waves in the m iddle of the flat are available. As for the
sedim ent behaviour, it has been seen th a t tidal processes generally generate on shore
sedim ent tra n sp o rt an d induce accretionary convex profiles, whereas waves generate
erosional concave profiles. W e think th a t any residual equilibrium is som ehow virtual
and is actually a dynam ical one, any flat experiencing periods of tidal accretion and
wave erosion, often according to a seasonal cycle (e.g. K irby et ah, 1993). This frequent
com petition betw een tidal currents an d waves explains why observed tidal flat profiles
are n o t consistent w ith uniform ity of m axim um velocity or shear stress or dissipation
rate. Effectively, neither com p u tatio n s n o r m easurem ents conducted on the three
investigated sites show ed any uniform ity across the flats.
8. Conclusion
H ydrodynam ics of intertid al flats result from the com bination of several forcings,
m ainly the tidal currents, the waves an d associated winds, and drainage.
Tidal currents can be split into cross-shore an d long-shore com ponents. T he form er
is due to filling an d em ptying of the em baym ent over the flat. If the b o tto m slope is
regular, the m axim um value of this cross-shore flow is uniform and inversely p ro p o r­
tional to the slope in the low er flat, an d decreases in the upper flat. The long-shore
com ponent can develop w hen the flat is connected to an estuary and is n o t bounded
by headlands; across the flat, this long-shore com ponent is roughly p ro p o rtio n al to
the square ro o t of the w ater depth.
W ave m easurem ents on gently sloping flats indicate th a t wave heights are lim ited
by a fraction of the w ater depth, w hatever be the incident wave height, because of
1456
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
wave energy dissipation; the im portance of this effect is quantified by the nondim ensional param eter f w/ß (ratio betw een the wave friction factor and the bo tto m
slope). This local satu ratio n of wave height can be of practical use for m odelling, as it
gives an idea of the m axim um wave height related to a given forcing, w ithout running
a costly wave p ro p ag atio n m odel. In addition, the m axim um -induced b o tto m stress is
proved to occur at high w ater an d to be p ro p o rtio n a l to the m axim um w ater depth.
C om bined w ith m athem atical m odelling, observations on the three investigated
sites provided validations of these schem atic considerations. Besides, despite different
forcing ranges an d to p o g rap h ic configurations, m axim um velocities have been proved
sim ilar on m uddy bottom s, indicating som e relationship betw een the sedim ent nature
and tidal velocities.
C onsequences of tide an d wave forcings on sedim ent tran sp o rt have been listed.
Schem atically, tidal asym m etry induces onshore sedim ent tran sp o rt, then generates
accretion on the u pper flat leading to a convex b o tto m profile, w hich in tu rn favours
an ebb dom inance th a t enhances the seaw ards sedim ent tra n sp o rt etc. Finally a tidal
equilibrium is likely to occur, w ith a resulting slope and convexity. Such an equilib­
rium can be upset by wavy episodes th a t erode the flat, prevent deposition on the
upper flat an d favour offshore transport.
A m ong the other hydrodynam ical forcings, drainage is an im p o rtan t process, very
specific to tidal flats. Even if the induced shear stresses are n o t very high, drainage
contributes to the offshore tra n sp o rt of sedim ent th a t have been previously resusp­
ended an d freshly deposited on high w ater slack. C om bined w ith tidal currents,
drainage is p roposed to be responsible for the fo rm ation of longitudinal bedforms,
w hich often constitute a netw ork of parallel ridges and runnels located where the
slope is m inim um .
A cknowledgem ents
The au th o rs th a n k R. Silva Jacin to (IFR E M E R ) for helpful discussions on the
wave effects, H. Jestin (IFR E M E R ) for contrib u tin g to field experim ents, D. G ouleau,
J. G erm aneau, P -G S auriau an d S. R ob ert (C R E M A -L a Rochelle) for sharing their
know ledge on the B rouage m udflat an d W. V an D er Lee (IM A U , U trech t Univ.) for
com plem entary d ata on the Skeffiing m udflat. They also th an k K. D yer and other
reviewers for valuable com m ents on the m anuscript. The study has been partly
su p ported by the IN T R M U D project, w hich is funded by the C om m ission of the
E u ropean U nion, D irectorate G eneral for Science, R esearch and D evelopm ent, under
co n tract N ° M AS3-CT95-0022.
Appendix A. M axim u m tidal current on a flat with uniform bottom slope
Assum ing th a t the w ater elevation rem ains h o rizontal w hatever be the tidal phase,
the m agnitude of the cross-shore curren t can be estim ated by considering the conser­
vation of volum e as the w ater level rises an d falls (Friedrichs and Aubrey, 1996). If the
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1457
w ater level rises, the edge of the w etted area m oves lan dw ard by a distance determ ined
by the slope of the m udflat. T he volum e of additio n al w ater th a t m ust pass through
a plane (at location x) parallel to the shore depends on the rise in w ater level
m ultiplied by the distance from the plane to the w ater’s edge.
Friedrichs an d A ubrey (1996) show th a t the cross-shore current speed is given by
where xf (t) is the location of the w ater’s edge at tim e t, referred to as the tidal front by
F riedrichs an d A ubrey (1996) an d others. If the slope of the m udflat is approxim ately
uniform , the first term on the right-hand side of Eq. (1) is equal to the inverse of the
m udflat slope ß.
Assum ing a sinusoidal variation of the w ater level z, its rate of change can be
derived as
dz
kR
¿7
=
T
o r r tid e
( 2nt \
C° \\ T
)’
r tid e /
,4
( • )
where R is the tidal range an d T tide is the tidal period. Substituting for the rate of
change of w ater level from Eq. (2), Eq. (1) becom es
u =
1n R
( 2nt \
— eos - —
P1
for x < xf(t).
(A.3)
\ 1 t id e /
t id e
T hus for every location of the flat w hich is covered w hen the rate of w ater level
variation is m axim um (i.e. the section of the flat below the m ean w ater level), the
m axim um cross-shore current is approxim ately [cosine = 1 in (A.3)]:
“ m ax =
P1
•
(A -4 )
tid e
In the upper p a rt of the flat (above m ean sea level), Friedrichs and A ubrey (1996) show
th a t the m axim um current occurs a t the tidal front, and the intensity is given by
dxf
»max = —r dt
1 dz
7iR
(
. (2ß
\\
nR
(2ß
= w T -=
— eos arcsin — x f
= —- — / I — — .
f JJ
ß d t ß T liie
\
\R
ß T tiie V
\R
(A. 5)
w ith xf = 0 at m id-tide.
Appendix B. W ave height distribution on a dissipative flat with uniform bottom slope
Let us consider the cross-shore conservation of wave energy in steady state, w ithout
wave refraction, n o r generation
_d
dx
(E(x)Cg(x)) = — D(x),
(B.l)
P. Le Hir et al. / Continental Shelf Research 20 (2000) 1433-1459
1458
where C g is the wave group velocity, D the dissipation by b o tto m friction and E the
to tal wave energy, th a t can be evaluated as
E = l p g H 2,
(B.2)
where p is the w ater density and g the acceleration due to gravity. In the case of
a m onoch ro m atic wave, D equals the w ork of a q u ad ratic b o tto m friction at the speed
of the orbital velocity during one wave period, th a t is
D=^J(i^wM
wlM
wl)M
wdt’
(B-3)
where T is the wave period, mw the orbital velocity above the boundary layer and
/ w a wave friction factor. The term between parentheses represents the wave-induced shear
stress. In the case of a sinusoidal wave with a m axim um velocity U w, D can be integrated
D = ^ p f wU l .
(B.4)
In shallow w aters, C g = (gh)1/2, an d U w can be ap proxim ated by
Eq. (5) is valid for h « ( 3 g / 4 n 2) T 2 ^ 0 .7 5 T 2.
Finally, the energy conservation reduces to
f ( H 2hll2) = - ^ H
dx
371
3/ r 3/2.
(B.6)
In the case of a uniform slope, h = ßx, w ith x = 0 a t the shore b u t > 0 seaw ard, then
Eq. (6) becom es
— (H2xx 112) = — f- i t f 3x ~ 3/2dx
3n ß
(B.7)
R eferences
Anderson, F.E., 1981. The N orthern m uddy intertidal: seasonal factors controlling erosion and deposition
— A review. C anadian Journal of Fisheries and Aquatic Science 40 (Suppl. 1), 143-159.
Anderson, F.E., Howell, B.A., 1984. Dewatering of an unvegetated muddy tidal flat during exposure:
dessication or drainage? Estuaries 7-3, 225-232.
Bassoullet, Ph., Le Hir, P., Gouleau, D., Robert, S., 2000. Sediment transport on an intertidal mudflat: field
investigations and estim ation of fluxes within the Baie de M arennes-O léron (France). 20 (12/13),
1635-1653.
Brenon, E, Le Hir, P., 1999. Modelling the turbidity maximum in the Seine estuary (France): identification
of form ation processes. Estuarine, Coastal and Shelf Science 49, 525-544.
Christie, M.C., Dyer, K.R., Turner, P., 1999. Sediment flux and bed level m easurements from a m acrotidal
mudflat. Estuarine, Coastal and Shelf Science 49, 667-688.
P. Le Hir et
al.
¡ Continental Shelf Research 20 (2000) 1433-1459
1459
Collins, M.B., Ke, X., Gao, S., 1998. Tidally-induced flow structure over intertidal flats. Estuarine, Coastal
and Shelf Science 46, 233-250.
D ronkers, J., 1986. Tidal asymmetry and estuarine morphology. N etherlands Journal of Sea Research 20
(2/3), 117-131.
Dyer, K.R., Christie, M.C., W right, E.W., 2000. U le classification of intertidal m udflats 20 (10/11),
1037-1038.
Eisma, D., 1997. Intertidal Deposits: River M ouths, Tidal Flats and Coastal Lagoons. M arine Science
Series, CRC Press, Boca Raton, 507 p.
Friedrichs, C.T., Aubrey, D.G., 1996. Uniform bottom shear stress and equilibrium hypsom etry of intertidal
flats. In: Pattiaratchi, C. (Ed.), Mixing in Estuaries and Coastal Seas, Coastal and Estuarine Studies, vol.
50. American Geophysical Union, W ashington DC, pp. 405-429.
Friedrichs, C.T., M adsen, O.S., 1992. N onlinear diffusion of the tidal signal in frictionally dom inated
embayments. Journal of Geophysical Research 97, N°C4, 5637-5650.
Guarini, J.M., Blanchard, G.F., Gros, Ph., Harrison, S.J., 1997. Modelling the m ud surface tem perature on
intertidal flats to investigate the spatio-tem poral dynamics of the benthic m icroalgal photosynthetic
capacity. M arine Ecology progress Series 153, 25-36.
Kirby, R., Bleakley, R.J., W eatherup, S.T.C., Raven, P.J., D onaldson, N.D., 1993. Effect of episodic events
on tidal m ud flat stability, Ardmillan Bay, Strangford Lough, N orthern ireland. In: M ehta, A.J. (Ed.),
N earshore and Estuarine Cohesive Sediment T ransport, Coastal and Estuarine Studies N o 42, AGU,
pp. 378-392.
Lee, S.C., M ehta, A.J., 1997. Problem s in characterizing dynamics of m ud shore profiles. ASCE Journal of
Hydraulic Engineering 123 (4), 351-361.
Li, Y., M ehta, A.J., 1997. In: Burt, N., Parker, R., W atts, J. (Eds.), M ud fluidization by w ater waves.
Cohesive Sediments. Wiley, New York, pp. 341-352.
Pethick, J.S., 1980. Velocity surges and asymmetry in tidal channels. Estuarine and Coastal M arine Science
11, 331-345.
Postm a, H., 1961. T ransport and accum ulation of suspended m atter in the D utch W adden Sea. Netherland
Journal of Sea Research 1, 148-190.
Roberts, W., Le Hir, P., W hitehouse, R.J.S., 2000. Investigation using simple m athem atical models of the
effect of tidal currents and waves on the profile shape of intertidal m udflats 20 (10/11), 1079-1097.
Roberts, W., W hitehouse, R.J.S., 2000. Predicting the profile of intertidal mudflats formed by cross-shore
tidal currents. Proceedings of IN T E R C O H ’98, Korea, in press.
Silva Jacinto R., Bessineton, Ch., Levoy, F., Védieu, C.H., Lesourd, S., Rousset, H., Benoît, L., Jestin, H.,
M onfort, O., 1998. Réponse de la vasière N ord aux forçages météo-océaniques. R apport final du
thème Hydrodynam ique et T ransport Sédimentaire, program m e scientifique Seine Aval, avril 1998,
pp. 99-111.
Soulsby, R.L., Hamm, L., Klopm an, G., M yrhaug, D., Simons, R.R., Thomas, G.P., 1993. W ave-current
interaction within and outside the bottom boundary layer. Coastal Engineering 21, 41-69.
Uncles, R.J., 1981. A note on tidal asymmetry in the sevem estuary. Estuarine, Coastal and Shelf Science 13,
419-432.
Wells, J.T., Kemp, G.P., 1986. Interaction of surface waves and cohesive sediments: field observations and
geologic significance. In: M ehta, A.J. (Ed.), Estuarine Cohesive Sediment Dynamics, Lecture notes on
Coastal and Estuarine Studies N°14. Springer, Berlin, pp. 43-65.
W hitehouse, R.J.S, Bassoullet, P., Dyer, K.R., Mitchener, H.J., Roberts, W., 2000. The influence of bedforms
on flow and sediment transport over intertidal mudflats. Continental Shelf Research.
W hitehouse, R.J.S., Mitchener, H.J., 1998. Observations of the m orphodynam ic behaviour of an intertidal
m udflat at different timescales. In: Black, K.S., Paterson, D.M., Cramp, A. (eds) Sedimentary Processes
in the Intertidal Zone. Geological Society, London, Special Publication 139, pp. 255-271.