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Netherlands Journal of Sea Research

20 (2/3): 117-131 (1986)

117

TIDAL ASYMMETRY AND ESTUARINE MORPHOLOGY

J.DRONKERS

Divisionof TidalWaters,Rijkswaterstaat,v. Alkemadelaan400,2597AT TheHague,TheNetherlands

ABSTRACT

Estuarine morphology is to a large extent deter-mined by the residual sediment transport pattern.However, the inverse statement is also true.Residual sediment transport depends on dif-ferences in magnitude and duration between ebband flood tidal currents. Such differences ("tidalasymmetry") are produced by the distortion ofthe tidal wave propagating on the coastal shelfand entering bays and estuaries. In this study therelationship between tidal asymmetry and estu-arine morphology is investigated. Based on theo-retical considerations some general principlesare derived and compared with fieldobservations.

1. INTRODUCTION

The evolution of an estuary depends essentially ontwo processes:

-the long-term averaged sediment supply from in-land or coastal origin, and the direction andmagnitude of the long-term averaged sedimenttransport,

- abrupt changes in the estuarine morphologycaused by storm surges or by engineering works.

The present study is concerned with the first pro-cess. The second process will be invoked whensome practical examples are discussed.

It is often conjectured that estuaries tend to fill inwith sediment and will, therefore, ultimately disap-pear. In the final stage of evolution, river water isdischarged directly on the coastal shelf (SCHUBEL&MEADE, 1977). However, temporarily certainestuaries may behave as erosion basins, as a conse-quence of externally induced changes in sedimentsupply or flow regime. Whether stable estuaries exist(sediment inflow on the average balanced by sedi-ment outflow) is not known, and it also seems anacademic question: there always exist external con-

ditions (e.g. the mean sea level) subject to variations(POSTMA,1980).

The sediment supply and sediment transport pat-tern in an estuary depend on several factors, someof which will be discussed in more detail:

-(a) The river inflow, with related to it (a1) an inputof sediment and (a2) a cross-sectional flow structureaffected by density differences. Density differencestend to increase flood currents in the deepest parts ofthe channel, in particular in the lower half of the ver-tical, and tend to decrease flood currents near thesurface and in the shallow parts of the cross-section.The inverse holds for the ebb currents.

This flow structure has an important impact on thesediment transport. It has been discussed already bymany authors, for example POSTMA(1967), FESTA&HANsEN(1978), ODD & OWEN(1972), ALLENet al.(1977), ARIATHURAIet al. (1977). The present studywill not enter further into details.

-(b) The sediment characteristics. In general awide spectrum of sediment types is present insuspension at the same time. In this study only acrude distinction between "fine" and "coarse" sedi-ment will be made, based mainly on fall velocity w(for "coarse" sediment w ~ 10-1 m.s -1, for finesediment W:$ 10 - 2 m.s - 1, see also section 2). Forboth sediment types two transport modes exist:along the bottom (coarse grains moving by tractionin the bedload, fine sediment moving as fluid mud),or as suspended load. As will be argued in section 2,tidal asymmetry affects suspension transport in par-ticular. The influence of tidal asymmetry on theresidual fluxes of coarse and fine sediment is dif.ferent, owing to different transport properties: thesuspension load of coarse sediment is stronglylimited by the current speed and it adapts to changesin the current speed rapidly in comparison with thetidal time scale. For fine sediment, saturation of thesuspended load seldom occurs. Most fine sedimentsettles only at very Iow current speed and the settlingtime delay is important.

118 J. DRONKERS

-(c) Wind waves, swell.The purelywave-inducedsediment transport is relatively less important inestuaries than on the coastal shelf. However, the in-fluence of wind waves, in tidal flat areas at highwater, on the resuspension of sediment is a signifi-cant phenomenon (ANDERSON,1972; McDoWELL&O'CONNOR,1977). In combination with subsequenttidal transport, it can cause a substantial seawardsediment flux.

-(d) The current velocity distribution and in particu-lar its variation during a tidal cycle. Twoelements aremainly responsible for residual sediment fluxes (inaddition to density influences):

-(d1)The tidal variation at the sea entrance, whichbears the characteristics of tidal wavepropagation onthe coastal shelf (amplification, distortion),

-(d2) The tidal propagation inside the tidal basin,generally consisting of a complex geometricalsystem formed by meandering and braiding channelsand tidal flats.

The present study is mainly devoted to aspect (d): theanalysis of tidal wave deformation in shallow systemswith a regular or a complex geometry, and its impacton the residual sediment flux. Globally speaking thefollowing features of tidal wave deformation are rele-vant for residual sediment transport (a more detaileddiscussion is presented in section 2):

- a difference between the maximum tidal currentsduring ebb and flood, which in particular affects theresidual flux of the coarse suspended fraction,

- a difference between the slack water periods pre-ceding ebb and flood, which particularly influencesthe residual flux of the fine suspended fraction.

In sections 3 and 4 it will be shown that a tidal wavechanges its original sinusoIdal shape and becomesasymmetric when it enters a tidal basin:

- if the mean water depth is so small that the tidalvariation cannot be neglected; the geometry changessignificantly with tidal level, or

- if the velocity field changes significantly over adistance comparable with the tidal excursion (forexample, the channel geometry varies over suchdistances).

In addition, the sinuso'idal shape of a tidal wave isaltered by the quadratic dependence of bottom fricti-on on the current speed. However,this quadratic cha-racter conserves the ebb-flood symmetry, and no re-sidual sediment flux results. Also, the main characte-ristics of tidal distortion will be demonstrated on asimplified model with bottom friction depending li-nearly on the current velocity. In the appendix, firstorder approximations of the tidal wave deformationdue to different non linear terms are presented for a

uniform rectangular basin with and without wavereflection.

Section 2 deals with the influence of tidal asymme-try on the residual sediment flux. In sections 3 and 4,the propagation and distortion of a tidal wave in diffe-rent types of basins is discussed: tidal rivers with nosubstantial wave reflection (section 3) and almost co-oscillating short tidal basins with a complex geome-try (section 4). In section 5 field observations of theresidual sediment flux in two tidal basins in the Ne-therlands are discussed as an example. In section 6the principal results are briefly summarized.

Many of the ideas today forming the foundation ofthe geomorphological science of estuarine andcoastal environments have been developed by HenkPostma. His sciehtific work has been an importantsource of inspiration for this study.

Acknowledgements.-Numerical model simulationswere performed by M. Geurtz in order to check thegeneral validity of several statements on the relation-ship between tidal asymmetry and estuarine mor-phology. Improvements of the draft have been sug-gested by J.R. van der Berg, L. Kohsiek and C.Louisse. The figures were prepared by Th. Vogel andthe typewriting by Mrs E. Goldbach.

2. SEDIMENT TRANSPORT DUE TOTIDAL ASYMMETRY

This section contains some general considerationson the residual flux of sediment in tidal environments.Successively, bed load transport and suspensiontransport will be discussed for both coarse and finesediment.

Bed load transport of coarse sediment consistsof grains with a diameter in the order of 200 J.lmorlarger, rolling or jumping over the bed (YALlN,1972;McDoWELL& O'CONNOR,1977;HEATHERSHAW,1981).The velocity of such grains is much smaller than thecurrent speed. As a consequence the local dominan-ce of ebb or flood currents mainly determines the re-sidual bed load flux. A general feature of sandy baysand estuaries is the presence of a pattern of alterna-ting ebb and flood channels (VANVEEN,1950; ROBIN-SON, 1960). As stated by LAMBIASE(1980), slow-moving grains transported by traction are trapped insuch channel systems. (On the contrary, suspendedgrains generally movefast enough to bypass channelsections where locally the residual current is oppo-sed). The conclusion is that in most sandy estuariesan ebb-flood asymmetry of the tidal wave does notgreatly affect the residual flux of bed load sediment.

Transport of fine sediment along the bottom is ge-

TIDAL ASYMMETRY AND ESTUARINE MORPHOLOGY

nerally described as "fluid mud". This phenomenonhas been observed in severalestuaries (e.g.the Tha-mes and the Gironde) and is associated with the oc-currence of a turbidity maximum (INGLlS& ALLEN,1957;ALLENet al., 1977).In these estuaries a distinctebb-flood asymmetry in the near-bottom layer is cau-sed by estuarine gravitational circulation. Again, anadditional ebb-flood asymmetry due to tidal wavedis-tortion is not the major factor for residual transport,although it may have a quantitatively significantinfluence.

Suspension transport of coarse sediment posses-ses the following characteristics: the maximum sedi-ment load in suspension at a given current speed islimited and depends on particle size and density(BRUUN,1978;DYER,1980).The saturation load is ra-pidly reached (as compared to the tidal time scale).Commonly found sediment particles with thesetransport characteristics in the estuarine environ-ment are fine sands (size on the order of 100j.lm)andlarge biologically bound aggregates (size on theorder of a few hundred j.lm) (BIGGS, 1978; POSTMA,1980; EISMAet al., 1980). The saturation load in-creases very strongly with increasing current veloci-ty. Therefore, a tidal asymmetry in the current veloci-ty variation mayaffect the residual transport of coarsesediment. The most significant asymmetry in thisrespect is a difference between the maximum currentvelocities occurring during flood and ebb. For exam-ple, if flood velocities exceed ebb velocities in com-pensation of a longer ebb duration, then a residuallandward transport of coarse sediment will exist.

In many estuaries, the suspended load is mainlycomposed of fine sediment: particles with a size bet-ween 1 and 10 j.lm (quartz, feldspar, clay minerals)and aggregates with a size up to 100 j.lm (MEADE,1972; EISMAet al., 1980). The transport of thismaterial is more complicated than the transport ofcoarse sediment. This is due in particular to thecohesive properties (consolidation and flocculation)and related time lag effects. Even at rather small cur-rent velocities (a few dm.s -1), an important load offine sediment can be maintained in suspension. Ac-tually, saturation seldom occurs. The concentrationof fine suspended sediment is limited by the availabil-ity of erodible bottom material, i.e. by consolidation orby the presence of an overlying coarse sedimentlayer. Weakly consolidated fine sediment isresuspended when the current velocity reaches acritical value Ue (see, for example, CREUTZBERG&POSTMA,1979; MEHTA & PARTHENIADES,1982;DRONKERS,1985). When the current velocity in-creases further, more consolidated fine sediment anddeeper layers may be brought in suspension.

119

Part of the fine sediment load settles in the periodaround slack water, in particular the aggregates witha size on the order of 100 j.lm and larger (POSTMA,1960; CHASE,1980).Adhesion on the bottom may oc-cur when the current speed has dropped below athreshold value Ud,Ud< ue (EINSTEIN& KRONE,1962).It follows that the fine sediment load responds morestrongly to tidal variations in the period around slackwater than around maximum current speed. Theslack water periods will be designated as SBE andSBF: Slack tide Before Ebb (seawardflow) and Slacktide Before Flood (landward flow), respectively.

For the residual transport of fine suspended sedi-ment, an approximate analytical expression hasbeen derived by DRONKERS(1985),based on a quali-tative description by POSTMA(1961).In this derivationthe excursion of sediment particles through the estu-ary is followed during a tidal cycle. Relevant quanti-ties are the water depth and the current velocity vari-ation, which are compared at t+ (=SBE) at a locati-on x+ (=the landward limit of the tidal excursion),and at t- (=SBF) at a location x- (=the seaward li-mit). The resulting expression for the net transportedsediment mass through a cross section x =='12(x+ + x-) during a tidal cycle reads:

M=j.I+).+ -j.l-).- (1)

where j.I+ (respectively j.I-) is the amountof sedi-ment settled on the bottom at x + (respectively x-) inthe period of SBE (respectively SBF), and). +(respectively). -) is the distance travelled by fluidparcels in the SBE (respectively SBF) time interval~t + (respectively ~t-) during which the fine fractionremainssettled.The quantitiesj.I::!:and),:!: can beevaluated from the approximate expressions:

j.I:!:(x)==U):!:(x).max. susp. conc. (x)

M :!:(x:!:)w:!:(x)==As(x:!:, t:!:)[1-exp(-w d )1 (2)

h(x:!:, t:!:)

w = fall velocity,As = stream cross-section, ~td = timeinterval for which I u I < ud'

).:!: (x) == 1!2ue.~t:!: (x:!:) (3)

The physics behind these equations is obvious: theamount of sediment j.I+, which is settled per unitlength in the period around SBE, will not follow thetidal motion before the ebb current reaches the criti-

cal speed for erosion ue' In this lapse of time the

120 J. DRONKERS

settled sediment is displaced with respect to thesuspended sediment in landward direction overa dis-tance which on the average equals A+. Around SBFa similar relative displacement will occur of a sedi-ment mass J.I- in seaward direction over an averagedistance A-.

The equations (2) and (3) show that a landward re-sidual flux of fine sediment is favoured if:

- the channel depth decreases in landward direc-tion, h(x+, t+)<h(x-, t-),

- the velocity variation is slower around SBE thanaround SBF,

I du/dt I + + < I du/dt I - -x ,t x . t

This last condition can be realized either by a distorti-on of the tidal wave (see the sections 3 and 4) or bya landward decrease of the current velocity. The lat-ter aspect has been demonstrated in particular byVAN STRAATEN & KUENEN (1959).

Finally the influence of wind wavesshould be men-tioned. In addition to tidal currents, short waves alsocontribute to bringing and keeping sediment insuspension. Wave energy dissipation is mainlyrestricted to shallow regions, with water depth in theorder of a few meters or less. In estuaries with land-ward decreasing depth, the amount of sediment de-posited in the period around SBE is more stronglydecreased by waveaction than the amount depositedaround SBF. In such estuaries wind wavescounteracta landward residual transport of fine sediment by tidalcurrents, or enhance a seaward residual transport.

The seaward flux due to wind waves can be parti-cularly important in estuaries with high tidal flats (si-tuated above mean sea level). In a meandering chan-nel systemsuch high tidal flats are built up by a near-bottom flow component which is directed up slope atthe headlands during both ebb and flood (HEATHER-SHAW& HAMMOND,1980). The fine sediment erodedfrom the tidal flats by waves in the period around highwater remains in suspension during most of the ebbperiod and is, therefore, transported a long distancein seaward direction. Coarse sediment is alsobrought in suspension by wind waves. It will be depo-sited sooner than the fine sediment and will thus betransported less far by the ebb current. However,thiscoarse sediment is not so easily resuspended by theflood current: in tidal flat estuaries wind waves alsocause a net seawardtransport of coarse sediment. Inother words: tidal flat formation by tidal currents,counteracted by wind wavesmay finally lead to an ex-port of sediment.

3. DISTORTIONOF A NON-REFLECTEDTIDAL WAVE

The cross-sectionally integrated tidal equations, validfor wide shallow basins (b:ph), read:

~ +! ~ (Asu)=Odt b dx(4)

<!.':'+u<!.':'+g~+F~ =0dt dx dx h+~

(5)

Here the total cross-section A (width at the watersurface=b) has been divided into two parts. The enti-re transport takes place in one part (crosssection=As' average depth=h). The other part(which is shallow, the depth averaged velocity beingless than, about, 0.3u) is considered to act as a stora-ge area only. The friction term is not quadratic but li-near with the current velocity. As explained, essen-tially this simplification does not affect the ebb-floodasymmetry of the tidal wave.Typicalvalues of the fric-tion coefficient F range between 10-3 and 5.10-3m.s-1.

In this section long inshore tidal basins or tidal ri-vers are considered in which the tidal wave can pro-pagate without substantial reflection. In practice, thisimplies a geometry without important cross-sectionalvariations. If a sufficiently regular geometry is assu-med, the equations (1) and (2) can be simplified tothe equations (A1)and (A2)of the appendix. In theseequations three non-linear terms are responsible forthe distortion of the tidal wave.The influence of theseterms is demonstrated in the appendix and the Figs10a to c. A qualitative discussion is presented below.

In the continuity equation (A1)the term ~du/dx re-presents the tidal variation of water depth. On theaverage the water depth is larger during flood thanduring ebb.Therefore, a residualdischarge in the wa-ve propagation direction will exist unless the floodcurrents are smaller than the ebb currents. This phe-nomenon is usually described as "Stokes drift".

In the momentum equation (A2) the term udu/dxconstitutes a contribution to the acceleration duldt,which tends to increase the magnitude of the accele-ration during flood and to decrease the magnitudeduring ebb. The result is contrary to the Stokes drift:a relatively shorter flood period with relatively highervelocities, as compared to the ebb period. However,Stokes drift is the dominating effect, see Fig. 10a.

A very important aspect of the non-linear termsudMdx and udu/dx is to introduce total time derivati-ves d/dt instead of partial derivates d/dt,d/dt= d/dt+ud/dx. For an observer moving with the

TIDAL ASYMMETRY AND ESTUARINE MORPHOLOGY

current velocity u the tidal equations remain linear(except for the term ~du/dx).As a consequence, thewave crest and wave trough propagate at differentvelocities, c + umaxand c - umaxrespectively. Herec=~ and umax== ac/h.Theterm~du/dxyieldsa lar-ger water depth for wave propagation in the crestthan in the trough V'g(h+ a) and V'g(h- a) respec-tively. Combining both effects the wave crest moves

approximately with velocity ~ (1+ ~~) and the

through with velocity ~ (1 - ~~ ). As a result, the

rising part of the wave surface becomes steadilysteeper, and the falling part steadily flatter, see Fig.10a. This result was established long ago by AIRY(1842). Ultimately this deformation leads to a tidalbore and wave breaking.

A tidal bore develops in an estuary if the tidal rangeb.~is sufficiently large and if the bottom slope Ib inthe zone situated around mean coastal sea level issufficiently gentle (COMOY,1881; LYNCH,1982)

~~:z: ~lb'T..J9(h+ <~»

See Fig. 1a. Well-known examples are the riversTSientang,Severn and Hooghly.

In most tidal rivers this final stage of distortion isnot reached. In general only a slightly steeper risethan decrease of water levels is observed. According-ly, the slack water period preceding flood is longerthan the slack water period preceding ebb. Examplesare shown in Fig. 2. As discussed in section 2, importof fine sediment is favoured by this tidal asymmetryover export.

From the analysis of IWAGAKI& SAKAI(1974)it fol-lows that the above discussion of wave deformation

(J

"EAN SEA LEvELTIDAL FLAT

c) RISING TIDE

.- - - - -~ - - H-'-1_LLlLLfIif'i

~./1//11/1111/1/1/1' m

b) FALLING TIDE

Fig. 1.Tidal propagation on a gently sloping tidal flat. a. for-mation of a bore during rising tide. b. Development of a

strong surface inclination during ebb.

121

TIDELEVEL

Iml j

CURRENT

VELOCITY [rn/s ]

t

.,OFLOOD

.0.5

0EBB

-0.5T

.1.0

f\/ ~

'2

.1 .0.5

0 0

-1 -0.5

-2 -1.0/-1.5

.2 ".0FLOOD

.0.5'1

0 0

-1

-2

-3 -1.5/.2.0

.3 '15

-2 01.0

.1 .0.5

0 0

-1 -0.5

-2 -1.0

- 3 '1.0FLOOD

.O.~

0

-0.5EBB

-1.016 [hours J0 4 B 12

fig. 2. field observations of tidal wavepropagation in rivers.

Ib

by non-linear terms remains qualitatively valid if, in-stead of a constant depth, a slowly decreasing depthin the wave propagation direction is considered,which is in general more realistic for tidal rivers.

A particular aspect of the previous discussion isthe tidal wave distortion in shallow coastal seas.Along the western coast bf the Netherlands a semidi-urnal tidal wave propagates from south to north, be-longing to the amphidromic system in the SouthernBight of the North Sea. Along the northern coast ofthe Netherlands a semidiurnal tidal wave propagatesfrom west to east, belonging the amphidromic systemof the central North Sea. Both tidal waves are distor-ted when running along the Dutch coast, and on theway exhibit an increasingly stronger rise and slower

122 J. DRONKERS

fall, see Fig. 3. This has important consequences forthe asymmetryof the co-oscillating tides in the Dutchtidal basins and estuaries, as will be shown in secti-ons 4 and 5. The generation of higher harmonic tidalconstituents in coastal shelf seas and their influenceon sediment transport have also been discussed re-cently by PINGREEet al. (1984).

So far friction has been neglected. However,frictio-nal influences cannot be ignored in most coastalseas and estuaries. The non-linearity of the frictionterm causes, in the uniform systems consideredhere, a larger frictional influence at Iow water than athigh water,yielding (slightly) larger flood than ebb ve-locities (see Fig. 10b).Another aspect is the retardati-on of the tidal velocity with respect to the water levelelevation. The maximum flood velocity occurs duringrising tide, the maximum ebb velocity during falling ti-de; the magnitude depends on the actual inclinationof the water surface. The formerly discussed steeperrise of the tide and slower decrease caused by thenon-linear terms u6u/6x and 6u~/6xproduce, in com-bination with friction, a relatively larger flood velocityand a relatively smaller ebb velocity. These effectscounteract and even dominate the Stokes drift, asshown in Fig. 10c. The occurrence of larger maxi-

mum flood velocities than ebb velocities is a com-monly observed feature in tidal rivers (at least at Iowriver runoff), see Fig. 2. It tends to favour landwardtransport of sediment over seaward transport.

As mentioned in section 1, river runoff produces across-sectional distribution of current velocitieswhich enhances seawardtransport in the near surfa-ce part, but counteracts seaward transport in thedeeper parts of the channel. This phenomenon willnot be discussed here, but should be considered forpractical applications. Finally, it is noted that river run-off not only influences the cross-sectional distribu-tion of currents. A distortion of the tidal wave may alsoresult, at least if river and tidal discharges are com-parable (GODIN,1985). The main cause is the qua-dratic dependance of bottom shear stress on currentspeed (HEATH,1981).However,the direction of the re-sidual sediment transport is not strongly influenced:at high river runoff seaward transport dominates(MIGNIOT,1968).

4. DISTORTIONOF A STANDINGTIDAL WAVE

In basins with a length I much shorter than the tidalwavelength L or a constriction at a distance I from the

(m)

2

I

0

-,

-2

:V" A IT- , \..,J '\..,

:lP ~ ID-, ~.....

:~ r- Dl-1 ~

'p0 1\ 1L

-, V \jFig. 3. Distortion of the tidal wave propagating along the Dutch coast (RIJKSWATERSTAAT).

TIDAL ASYMMETRY AND ESTUARINE MORPHOLOGY 123

sea boundary, 1« L, the major part of the tidal waveISreflected at the head. The simplest schematizationISa rectangular basin of constant depth. In the ap-pendix the influence of the non-linear terms in thecontinuity equation and the momentum equation isdemonstrated for such a simple geometry.

Without the friction term the tidal motion is almosta standing wave, slightly distorted by the non-linearterms d~u/dxand udu/dx. This distortion consists es-sentially of a relatively longer slack water period befo-re ebb than before flood (Fig. 10d).The reason is thatchanges in water level propagate faster at high waterthan at Iowwater; a nearly horizontal water surface ismore readily reached and maintained. As a result thewater level inclination and corresponding flow acce-leration is larger in magnitude at Iow water than athigh water.

The influence of the friction term is twofold. Firstly,part of the energy of the incoming tidal wave is absor-bed; thus the reflected wave is smaller than the inco-ming wave: the tidal motion has the character of apartly progressive wave (lpPEN& HARLEMAN,1966).Secondly, the frictional influence is less around highwater than around Iowwater.The effect is opposite tothat of the other non-linear terms: around high water(just before and after slack water) the magnitude ofthe current is larger than around Iow water (see Fig.We)

I du/dt I SBE> I du/dt I SBF

In a tidal basin with landward decreasing depth theinfluence of friction is not the same when the tidal va-riation of the current velocity in a moving frame isconsidered. If the decrease in channel depth on theflood excursion of a fluid parcel exceeds the tidal ran-ge, then the friction experienced in the period aroundhigh water slack (SBE) is larger than the friction ex-perienced around Iow water slack (SBF).Consequently,

I du/dt I SBF> I du/dt I SBE

lor a fluid parcel moving with the tide: in a rectangu-lar basin with landward decreasing depth, a land-ward residual transport of fine sediment is favoured.

The distortion of a tidal wave in basins of limitedlength due to both non-linear propagation terms andfriction has been studied by several authors: KREISS(1957), HEATH(1981) and UNCLES(1981).They findthat the maximum flood velocity exceeds the maxi-mum ebb velocity, at least in the inner part of theestuary (in the absence of river discharge). The rea-son is again that water level changes propagate

faster around high water than around Iow water.Thetime delay between high water at different locationsin the estuary is shorter than the delay between Iowwater. Similarly the time delay between SBE at diffe-rent locations is smaller than the time delay betweenSBF (dt+/dx < dt -/dx). In other words, the flood du-ration in the inner part of the estuary is shorter thanthe ebb duration. Consequently, the maximum floodvelocity exceeds the maximum ebb velocity. This ef-fect is enhanced when frictional energy dissipation isimportant. In that case, water level variations are dif-fused through the estuary, rather than propagated.As shown by LE BLOND(1978),diffusion speed de-pends even more strongly on water depth than wavespeed.

In the case of short tidal basins, distortion by non-linear terms is rather small. The water motion has es-sentially a co-oscillating character with nearlyconstant phase and water-levelthroughout the basin.In fact the most important causes for tidal asymmetryare (see also the studies of PETHICK,1980; BOON&BVRNE, 1981):

- an asymmetry in the tidal boundary condition,- a variation in the basin geometry with water-level.In a co-oscillating tidal basin the current velocity

can be obtained by integration of the continuity equa-tion (1):

E ~Iu==A dt I x=os

(6)

Here ~ = JxlbdX is the storage surface of the basin.This quantity, as well as the stream cross section As,depends on the water level ~.

The influence of the tidal boundary condition ~(t)on the current velocity follows directly from equation(6). As mentioned already, a distortion of the tidal wa-ve propagating over the shallow coastal shelf influen-ces the tidal motion in the adjacent estuaries. Forexample, the relatively fast rise and slow decrease ofthe tidal wave propagating along the Dutch coast isthe major cause of the occurrence of a greater maxi-mum current during flood than during ebb in theDutch tidal basins (see Fig. 4). The boundary tidealso affects the slack water periods before ebb andflood, but to a lesser degree. Here the major causeof asymmetry is the tidal variation of the basingeometry.

The time derivative at slack water of equation (6)yields:

du I ==£ /)2~ Idt I slack As dt2 I slack

(7)

124 J. DRONKERS

flotlEvEl

[m]

CURRENTVELOCITY

[m/si

'10FLOOD

.05

. ~

-0.5EBB

-10

. ,0 0

. ,

. 2

'15

. 2

-75

.10

.1 .0.5

0 0

.1 -0.5

- 2 -1.0DISCHARGE

7510' [m'fs]

.2 50

.1 25

0 0

- 1 -25

- 2 -50

75

.2 50

.1 25

0 0

- 1 -25

- 2 .50

0 8 12 16 {hours]4

Fig. 4. Tidal variation of discharge/current velocity (-) andwater level (- - -) in Dutch tidal basins (RIJKSWATERSTAAT,

unpublished data).

On the basis of this expression one may distinguishbetween two types of tidal basins:

(1)The relative increase of stream cross-section Aswith increasing water-level is smaller than the relativeincrease of storage surface. The corresponding tidalbasins havedeep channels (h>> a) and/or high tidalflat areas (on the average above mean sea level).This characteristic favours a longer slack water peri-od before flood than before ebb, I du/dt I SBF <I du/dt I SBE'Examples are shown in Fig. 5.

(2) The relative increase of stream cross-sectionwith increasing water-levelis greater than the relativeincrease of storage surface. The corresponding tidalbasins haveshallow channels (depth at most a few ti-mes larger than the tidal amplitude) and/or Iow tidalflat areas, which at high water become part of the

CURRENTVELOCITY

[m/s]

".0FLOOD

'0.5

[m/sI.1.0

FLOOD.0.5

EBB

DISCHARGE

[m'/s]

-0.5

-110'

-1.0[m'/sJ210'

1.10'

0

0

-0.5

0 8 11\ 20 thOU'>]4 12

Fig. 5. Field observation of geometry-induced tidal wavede-formation in type 1 tidal basins.

stream cross-section. This characteristic favours a

longer slack water period before ebb, I du/dt I SBE< I du/dt I SBF'Examples are shown in Fig. 6.

The above implies that the tidal variation of the wet-ted geometry affects the residual transport of fine se-diment. In the type 1 tidal basin, residual export of fi-ne sediment is favoured. In the type 2 tidal basins, re-sidual import of fine marine sediment can be expec-ted and retention of the fine fluvial sediment dischar-ged at the head.

It has been noted by SPEER& AUBREY(1985)andby BOON& BYRNE(1981)that large tidal flats may en-hance the maximum ebb current. The tidal wave pro-pagates faster in the channels than on the tidal flats.Therefore, the decrease in water-levelduring ebb ta-kes place later on the tidal flat than in the channel.This leads to an important water-level inclination (seeFig. 1b) and a strong current during the last stage ofthe ebb period. A net seawardflux of coarse suspen-ded sediment may result.

0

EBB-0.5

.1.5

.1.0

0.5

0

- 0 5

125

CURRENTVELOCITY

[mls]

5. TIDAL ASYMMETRY AND SEDIMENTTRANSPORT IN THE WADDEN SEA AND

THE EASTERN SCHELDT. 1.0

4 8 12 [hours]

In this section field observations in two tidal basins inthe Netherlands will be discussed as an example ofthe more general principles established previously. Inboth tidal basins fresh water inflow is extremelysmall. The current distribution is mostly tide-induced;wind forcing is a secondary effect.

The morphology of the Wadden Sea Ameland areaand the Eastern Scheldt basin is shown in Fig. 7, onthe same scale. Both tidal basins are much shorterthat the tidal wave length. In the Eastern Scheldt lar-ge deep channels bounded by levees lead to tidalflats situated in the landward part; in the Wadden Seaa sequence of large, medium and small channels ra-pidly lead from deep to shallow regions. According tothese morphological differences the tidal wave is dis-torted differently in both systems. This is most pro-nounced for the slack water periods. In Fig. 8 the ba-thymetry of the two tidal basins is represented by thecurves showing the surface at different depths. In theWadden Sea the relative increase of the volume with

.05

0

TIDAL ASYMMETRY AND ESTUARINE MORPHOLOGY

Fig. 6. Field observations of geometry-induced tidal wavedeformation in type 2 tidal basins.

OOSTERSCHELDE

0

~5::11

10 kmI

'

~.

::.'~-' ::>i ""-tf-,:.<4"-,~~~ \ j/? >~,'

'~I.~i~ft~~>:,r~~*~l':dWADDEN SEA AMELAND AREA

10m DEPTH LINE

. MEAN lOW WATER LINE

Fig. 7. Morphology of the Wadden Sea Ameland area and the Eastern Scheldt basin.

.05

0

- 0.5

. 1 0

.05

0

- 0.5

- 1.0

l0

126 J. DRONKERS

(0)

-"W

(m] --LW

( b)

[m]"W

II

_10,1

"W~~--- "-"-"-"-'\

\/

/.--/LW---

'.10' 10'

L . jb(')dX- - - WADDENSEA.AMELAND AREA- OOSTERSC"ELDE

BAT"YMETRY "" OOSTERSC"ELDE AND AMELAND TIDAL SYSTEMS,

(0) SURFACE AS A FUNCTION OF DEPT"

(b) RATIO OF STORAGE AREA ~ AND STREAM CROSS- SECTION A.

Fig. 8. Surface as a function of depth in the Wadden Sea

Ameland area and the Eastern Scheldt basin; the ratio "LIAsas a function of tidal level.

tidal level is greater than the relative increase of sur-face.The ratio '£./Asat highwateris smallerthan atIow water. According to equation (7) one may expect,in theWaddenSea Idu/dt ISBE< I du/dt I SBF'In theEastern Scheldt the opposite inequality should hold.Fig. 9 shows the agreement of these predictions withfield data.

The peak flood discharge slightly exceeds thepeak ebb discharge in both the Wadden Sea and theEastern Scheldt for an average tide, see Fig. 4. Theextent of tidal flat areas is insufficient to cancel theflood dominance due to the faster rise and slowerde-

crease of the tide along the Dutch coast (see Fig. 3).(Locally ebb currents may dominate flood currentsdue to the presence of ebb and flood dominatedchannels). Therefore, flood transport of coarse sedi-ment should in principle exceed ebb transport.

The tidal boundary curves (Fig. 3) further showthat the second derivative d2~/dt2presents a differentasymmetry in the Wadden Sea and in the EasternScheldt. In the Wadden Sea Id2~/dt21HW< I d2Vdt2I LW'This means that the distortion of thetidal wave running along the Dutch coast also contri-

U( t)

,[m"J

t [hOU," )

-,

WADDEN SEA AMELAND AREA LOCATION 8

U (t) \

[m/')

D

-,LAs

EASTERN SC"ELDT LOCATION WS

['O'm']Fig. 9. Current velocity variation in the Wadden Sea Ame-land area (location 8) and in the Eastern Scheldt basin (lo-

cation WS).

butes to the inequality Idu/dtISBE< I du/dt ISBF'Inthe Eastern Scheldt the boundarytide contributes tothe opposite inequality.

Largetidal flats are presentin the Wadden Sea andin the Eastern Scheldt. The majorpart is covered on-ly around high water. In this period bottom material(fine and coarse sediment)can be brought in suspen-sion and subsequently be transported by ebb cur-rents. Thus the residual landwardtransport by tidalcurrents of both coarse and finesediment is counter-acted. In the Wadden Seathe fine sediment fractionin the intertidal areas is much larger than in theEastern Scheldt. Therefore,one mayexpect that theinfluence of wind wavesaffectsmorestrongly the finesediment transport in the WaddenSea.

One may also expect that a residual landwardtransport of sediment prevailsunder calm weatherconditions, while seawardtransportdominates understorm conditions.

Field observations showthe following:In the Wad-den Sea, bottom soundingsovera periodof 20 yearsindicate an average bottomaccretionof 0.5-1cm.a-1(DE BOER& VISSER,1981).In the Eastern Scheldt,over the period 1872-1974an averageerosion of 1cm.a -1 is found, VANDENBERG(1984).However, thenet erosion rate has decreasedin the last decadeand does not exceed a few mm.a-1. An extensive

sediment transport measurementcampaigncoveringa whole year, but excluding stormperiods yielded anet export of sediment (essentiallyfine sediment)corresponding to an averageerosion of 0 to 3cm.a -1 (DRONKERS,1985).

TIDAL ASYMMETRY AND ESTUARINE MORPHOLOGY 127

TIDELEVEL

[m]

1.D

CURRENTVELOCITY

[m/s]

,.0

.0..0

.05

.0.0

-.0.5

-'.0 -,.0

::~ ~"

/-/' "-

.0..0 "-

-.0.5 ~2 .0..01..0 .05

.05

DDDD

-.0.5

-10 -0.5

.05

.0..0 DD

-.05

-,.0 -.05

1.0 .05

.0.5

DD .0..0

-.0.5

-1..0 -.05

Fig. 10. Time variation of water-level (- - -) and current

velocity (-) in a uniform channel with open sea boundary.

t = a cos.dt at x = O. The amplitude a = 1m, the depth

h = 10 m; for damped waces the friction coefficient F =0.002 m.s-1.a. Progressive non-linear wave without frictionat x= 100km.b. Progressivedampedwave,withouthnon-linear terms du~/du,udu/dx.c. Progressivedampednon-linear wave. d. Standing non-linear wave without friction atx = 25 km in a canal of length I = 50 km. e. Standing damped

wave, without non-linear terms duVdx, udu/dx.

The expression (1) for tide-induced residual trans-port of fine sediment yields a net import correspon-ding to an average bottom accretion of 1-6 cm.a-1for the Wadden Sea, and for the Eastern Scheldt anet export corresponding to an average bottom erosi-on of 0 to 0.5 mm.a -1.

A comparison of the field observations with thepredictions for fine sediment transport, taking intoaccount the qualitative tendencies that have been in-dicated for storm affects and for the residual trans-port of coarse sediment, allows the conclusion that:

- for fine sediment, the observed and predicted di-

rections of the residual fluxes are in agreement.- the magnitude of the residual flux of fine sedi-

ment is strongly influenced by wind waves, andstorms, especially in the Wadden Sea (see also WIN-KELMOLEN& VEENSTRA,1980; DRONKERS,1985)

- for coarse suspended sediment the landward di-rection of the tide-induced residual flux in the Wad-den Sea agrees with observations. In the EasternScheldt an important contribution to the residualtransport of coarse suspended sediment seems to beprovided by the action of wind waves in combinationwith tidal transport. The stronger impact of wind wa-ves in the Eastern Scheldt compared to the WaddenSea can also be traced to the different orientations ofthese basins with respect to the averagewind directi-on: in the Eastern Scheldt the direction of the mainchannel axis fits the average wind direction moreclosely.

The field observations indicate that a morphologi-cal equilibrium in the Wadden Sea and the EasternScheldt is not yet reached in the period under consi-deration. However, different processes involvingwind- and tide-induced processes yield mutuallycounteracting residual fluxes of the same order ofmagnitude. This implies that small morphologicalchanges can alter the total residual transport: the ac-tual statecannot be very far from morphological equi-librium. In fact, the major external change in theWadden Sea Ameland area during the past decadesis an increase of the mean sea levelof 0.2 cm.a-1 onthe average. If the sea level stabilizes, the most pro-bable evolution of the Wadden Sea is a further accre-tion of the tidal flats until a quasi-equilibrium betweentide- and wind-induced sediment fluxes is reached.

In the Rhine-Meuse-Scheldt delta successive engi-neering works have caused a considerable increaseof the tidal prism in the Eastern Scheldt during theperiod 1872-1974and in particular between 1965and1970. The tidal energy dissipation in the EasternScheldt increased. Erosion dominated oversedimen-tation and ebb sediment fluxes over flood sedimentfluxes, especially in periods shortly after man-induced alterations of the hydraulic regime. In the fu-ture a storm surge barrier in the mouth of the EasternScheldt, presently under construction, will decreaseagain the tidal influence and the sediment motion inthis basin. A slow infill with sediment is expected.

6. SUMMARY AND CONCLUSIONS

1)The most pertinent features of tidal wavedistortionfor residual sediment transport are:

- a difference between the slack water periodsbefore ebb and flood ( I du/dt I SBE*I du/dt I SBF)'

128 J. ORONKERS

whiCh affects in particular the residual transport ofthe fine fraction of the suspended load.

- a difference between the maximum currents du-

ring ebb and flood (u~bt"* uH6~~),which especially af-fects the residual transport of the coarse fraction ofthe suspended load.

2) In regularly shaped basins (no important widthvariation with water-level)and in the absence of riverinflow, the tidal wave tends to be distorted such thatu~;;d > u~b~x,I du/dt I SBE< I du/dt I SBF'This distor-tion is manifest in the inner part of both long and shorttidal basins (compared to the tidal wave length). It willcause a sediment infill of estuaries in periods of smallriver discharge.

3) In irregularly shaped estuaries (meandering andbraiding channel system, tidal flats) the tidal currentvariation is influenced by the geometry. Twotypes ofgeometry can be distinguished:

- shallow channels/landward decreasing depth, ti-dal flats below mean sea level.

- deep channels throughout, tidal flats abovemeansea level.

In the first case the slack water period before ebbwill exceed the slack water period before flood: a resi-dual import of fine sediment is favoured. In the se-cond case the inverse situation occurs.

In addition, large tidal flats cause an enhancementof the maximum ebb current and may, therefore, in-duce a seaward flux of coarse suspended sediment.

4) The distortion of the tidal wave propagating onthe coastal shelf also induces an ebb-flood asymme-try in the tidal current variation inside adjacent baysand estuaries. If the rise of the coastal water-level isfaster than the decrease, in a co-oscillating tidal ba-sin uH6~~>u~bt"and a sediment infill is favoured.

5) Wind induced resuspension of sediment mayalso present a tidal asymmetry. In many estuaries themajor part of the fine grained tidal flats are coveredaround high water only. In such cases wind inducedresuspension leads to an export of fine sediment.

6. REFERENCES

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ALLEN, G.P., G. SAUZAV, P. CASTAING & J.M. JOUANNEAU, 1977.

Transport and deposition of suspended sediment inthe Gironde estuary, France. In: M. WILEV. Estuarineprocesses Vol. 2. Academic Press N.V.: 63-81.

ANDERSON, F.E., 1972. Resuspension of estuarine sedi-ments by small amplitude wavesri. sedim. Petrol.42: 602-607.

ARIATHURAI,R., R.E. MACARTHUR& R.B. KRONE, 1977. Mathe-matical model of esturial sediment transport. Tech.Rep. 0-77-12,Environmental Effects Laboratory, U.S.Army Engineers Waterways Experiment Station: 1-70.

AVOINE,J., G.P.ALLEN,M. NICHOLS,J.C. SALOMON& C. LAR-SONNEUR,1981.Suspended-sediment transport in theSeine estuary, France: effect of man-made modificati-ons on estuary-shelf sedimentology, Mar. Geol. 40:119-137.

BERGVANDEN,J.R., 1984. Morphological changes of theebb-tidal delta of the Oosterschelde during recent de-cades, Geologie Mijnb. 63: 363-375

BIGGs,R.B., 1978.Coastal bays. In: RA DAVls.Coastal se-dimentary environments. Springer Verlag: 1-420.

BOER,M. DE& G.C.VISSER,1981.Erosie en sedimentatie inthe westelijke Waddenzee. Nota WWKZ- 80.HO01,Rijkswaterstaat, The Netherlands: 1-38.

BooN, J.D. & R.J. BVRNE,1981.On basin hypsometry andthe morphodynamic response of coastal inletsystems, Mar. Geol. 40: 27-48.

BRuuN, P., 1978.Stability of tidal inlets. Elsevier Amster-dam: 1-510.

CHASE,R.R.P., 1979.Settling behavior of natural aquat,icparticulates, Limnol. Oceanogr. 24: 417-426.

COMOV,M., 1881.Etude practique sur les marE~esfluvialeset la mascaret. Paris.

CREUTZBERG,F.& H. POSTMA,1979.An experimental appro-ach to the distribution of mud in the southern NorthSea, Neth. J. Sea Res. 13: 99-116.

DANKERs, N., M. BINSBERGEN, K. ZEGERS, R. LAANE & M. RUT-

GERS VANDER LOEFF, 1984. Transportation of water, par-

ticulate and dissolved organic and inorganic matterbetween a salt marsh and the Ems-Dollard estuary,the Netherlands, Estuar. coast. Shelf Sci. 19:143-165.

DRoNKERs,J., 1985.Tide-induced residual transport of finesediment. In: J. VANDEKREEKE.Proceedings of theSymposium on the physics of shallow bays and estua-ries, Miami 1984. Springer Verlag (in press).

OVER,K.R., 1980. Velocity profiles over a rippled bed andthe threshold of movement of sand, Estuar. coast.mar. Sci. 10: 181-199.

EINSTEIN,H.A & R.B. KRONE,1962.Experiments to determi-ne modes of cohesive transport in salt water, J. geop-hys. Res. 67: 1451-1461.

EISMA,D., J. KALF& M. VEENHUIS,1980. The formation ofsmall particles and aggregates in the Rhine estuary,....Neth. J. Sea Res. 14: 172-191.

FESTA,J.P.& D.V.HANSEN,1978.Turbidity maxima in partiallymixed estuaries: a two-dimensional numerical mo-del, Estuar. coast. mar. Sci. 7: 347-359.

'GODIN, G., 1985. Modification of river tides by the dis-charge, J. WatWay port coast. ocean Engng 111:257-274.

HANSEN,w., 1962. Tides. In: M.N. HILL.The Sea, Vol. I.John WHeyN.V.: 764-801.

HARLEMAN,D.R.F.,1971.One-dimensional models. In: Estu-arine modelling: an assessment. Tracor Inc., Austin,Texas: 50-51.

HEATH,R.A., 1980. Phase relations between the over- andfundamental-tides, Dt. hydrogr.Z. 33: 177-191.

HEATHERSHAW,AD., 1981. Comparisons of measured andpredicted sediment transport rates in tidal currents,....Mar. Geol. 42: 75-104.

HEATHERSHAW,AD. & D.C. HAMMOND,1980. Secondary cir-culations near sand banks and in coastal embay-ments, Dt. hydrogr. Z. 33: 135-151.

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INGLlS,C.C.&EH. AllEN, 1957.Theregimenof theThamesestuary as affected by currents, salinity and riverflow.--Proc. Inst. civ. Engng 7: 827-868.

IpPEN,A.T. & D.RF. HARlEMAN,1966. Tidal dynamics inestuaries. In: A.T. IpPEN.Estuary and coastline hydro-dynamics. McGraw-Hill,493-545 .

IWAGAKI,Y. & T. SAKAI,1972. Shoaling of finite amplitudelong waveson a beach of constant slop~Proc. Conf.coast. Engng 13 (1): 347-364.

KJERFVE,B., L.H. STEVENSON,J.A. PROEHl, T.H. CHRZANOWS-KI& W.M. KITCHENS,1981.Estimation of material fluxesin an estuarine cross section: a critical analysis of spa-tial measurement density and errors,.-Limnol. Ocea-nogr. 26: 325-335.

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LAMB,H., 1932. Hydrodynamics. Cambridge Univ. Press:1-738.

LAMBlASE,J.L., 1980.Hydraulic control of grain-size distribu-tions in a macrotidal estuary,.-Sedimentology 27:433-446.

LE BLOND,P.H., 1978.On tidal propagation in shallow ri-vers,.-J. geophys. Res. 83 (C9): 4717-4721.

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129

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130 J. DRONKERS

APPENDIX

The one-dimensional tidal equations (1-2)are solved for uniform systems (constant depth, width) to first order in a/h, accor-ding to standard methods as indicated, e.g. in LAMB(1932). For distinction between the impacts of the different non-linearterms, multiplication factors f1, f2 and f3 have been introduced. The equations read

cg+h~ +f d~u=0dt dx 1dx (A1)

du du d~ u . u~- +f u- +g-+F- -f F- =0dt 2 dx dx h 3 h (A2)

Solutions are presented for systemswith a prescribed water-levelvariation at the ocean boundary: W)=a cos at. The soluti-ons presented in HEATH(1980)for the same tidal equations (but with a quadratic friction law) satisfy different boundary con-ditions. The qualitative features of the solutions appear to be similar, however.

SOLUTIONS FOR PROGRESSIVE WAVES

1. Without friction (F=O), see Fig. 10a.

~= a[ cos(at-kx)-(f1 +2f2)i ~kX sin(2at-2 kX)]

u= ~ [cOS(at-kX)- ~ (1 + 1(2f2 -f,)cos(2at-2kx)+ 1(2f2 +f1)kx sin(2at-2 kX))]kh h 2 8 4

Here the wave number k is given by k = a/~. The validity of these solutions is restricted to

X<'<:h/ka(LAMB, 1932)

2. Non-linear friction (F> 0, f3= 1) but no other non-linear terms (f, = f2= 0), see Fig. 10b.

~= ~ k2R (1 - e2klX)+ 1a [ei(at- kx) - ~ (e2i(at - kx) - e2i(at - k'X))+c.c.]2h I k I 2 2 2h .

u= ~ [lei(at-kX) - ~ (le2i(at-kX)_1 e2i(at-k'X)) +c.c.]2h k 2h k k'

Here k and k' are complex wave numbers, kR=Re k>O, kl=lm k<O,

ala-if)k2= ~

gh '

2ala-if )

k'=~gh

3. Non-linear friction (F > 0, f3 = 1) and non-linear terms (f 1= f2 = 1), see Fig. 1Oc.

~=B(1-e2kiX)+~a[ ei(at-kx) +A( e2i(at-kx) -e2i(at-k'X)) +c.c.]

u=Ce2klX + aa [ lei(at-kX) + ((A- ~) 1e2i(at-kX)- ~ e2i(at-k'X)) +c.c.]2h k 4h2 k k'

Here A= - ~ (2 f + 3ia),2F h

a2a2 F kR

B= 4gh2 I k 12 (1-2fW 1<;)'C= - aa2kR

2 h2 I k I 2

TIDAL ASYMMETRY AND ESTUARINE MORPHOLOGY 131

SOLUTIONS FOR STANDING WAVES

A basin is considered with a closed end at x=1 and an ocean boundary at x=o.

4. Non-linear terms (11' 12",0), no Iriction (F = 0), see Fig. 10d.

a[

1 a2]~= - cos k(l-x) cos at-(I, +212)- 2 kx cos 2k(l-x)cos 2 atcos kl 8 h

u-- ~ [sin k(l-x)sin at-~ (~(2f2-ll)+ ~(212+1,)k[I+(I-X)Cot 2k(I-X)Dsin 2k(l-x)sin 2at]kh cos kl h 16 8

5. Non-linear Iriction (F>O, 13=1), no other non-linear terms (I, =12=0), see Fig. 10e.

Faa2 k2, cos 2kRI-cos 2kR(I-x) +k2R cosh 2k,l-cosh 2k,(I-x)~= .

4g h3 I k 12 kRk, cos 2kRI+cosh2k,1

+~a [COSk(l-x) eiat- a (COS2k(l-x)- cos 2kl cos 2k'(I-X»)e2iat+c.c.]2 cos kl 4h COS2kl cos 2k'l

u=i~ [Sin k(l-x) eiat- a (Sin 2k(l-x) - cos 2kl sin 2k'(I-X))e2iat+c.c.]2h k cos kl 4h cos2 kl k cos 2k'l k'