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Estuarine, Coastal and Shelf Science 79 (2008) 7–16

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Estuarine, Coastal and Shelf Science

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Estuarine, Coastal and Shelf Science

journal homepage: www.elsevier .com/locate/ecss

New analytical equation for dispersion in estuaries with a distinctebb-flood channel system

Anh Duc Nguyen a,b,d,*, Hubert H.G. Savenije a,b, Mick van der Wegen c, Dano Roelvink c

a Department of Management and Institutions, UNESCO-IHE Institute for Water Education, 7 Westvest, P.O. Box 3015, 2611 AX Delft, The Netherlandsb Department of Water Management, Faculty of Civil Engineering and Applied Geosciences, Delft University of Technology, Stevinweg 1, P.O. Box 5048,2600 GA Delft, The Netherlandsc Department of Water Engineering, UNESCO-IHE Institute for Water Education, 7 Westvest, P.O. Box 3015, 2611 AX Delft, The Netherlandsd Center of Hydroinformatics, Southern Institute for Water Resources Research, 2A Nguyen Bieu, Ho Chi Minh City, Vietnam

a r t i c l e i n f o

Article history:Received 6 August 2007Accepted 4 March 2008Available online 10 March 2008

Keywords:dispersionestuariesebb and flood channelstidal mixingtidal pumpingresidual circulationWestern Scheldt estuary

* Corresponding author. Department of ManagemeIHE Institute for Water Education, 7 Westvest, P.O. BNetherlands.

E-mail address: d.nguyen@unesco-ihe.org (A.D. Ng

0272-7714/$ – see front matter � 2008 Elsevier Ltd.doi:10.1016/j.ecss.2008.03.002

a b s t r a c t

Tidal pumping caused by residual horizontal circulation is an important but ill-understood mechanismproducing longitudinal salt dispersion in well-mixed estuaries. There are two types of residual circulationthat cause tidal pumping: (1) interaction of the tidal flow with a pronounced flood-ebb channel system;and (2) interaction of the tidal flow with an irregular bathymetry. Residual ebb-flood channel circulationis an important large-scale mixing mechanism for salinity intrusion, as shown in the Western Scheldt inthe Netherlands, which is a well-mixed estuary with a distinct ebb-flood channel system. This paperprovides a new simplified conceptual model and a new analytical equation for this type of mixing. Firstly,using a fully three-dimensional hydrodynamic model as a ‘‘virtual laboratory’’ and employing a de-composition method, the characteristics of the residual ebb-flood channel circulation in the WesternScheldt are analysed. Secondly, a conceptual model and an analytical equation determining the disper-sion coefficient are developed, which take into account relevant parameters for tidal pumping, such asthe tidal pumping efficiency, the tidal excursion and the length of the branches. Subsequently, the newlydeveloped equation is compared to the results of the ‘‘virtual laboratory’’. The comparison confirms anagreement between the newly developed equation and the ‘‘virtual laboratory’’ in determining the re-sidual transport and the tidal pumping dispersion coefficient. Finally, the equation is applied to obser-vations in the Western Scheldt. The application yields good results in determining the longitudinaldispersion compared to dispersion values obtained from the salt budget.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The tidal pumping mechanism, which is caused by residualhorizontal circulation, is an important but ill-understood part oftidal circulation producing longitudinal dispersion. Fischer et al.(1979) defined ‘‘tidal pumping’’ as the energy available in the tidethat drives steady circulations similar to what would happen ifpumps and pipes were installed to move water about in circuits.There are two types of residual circulation that cause tidal pump-ing: (1) interaction of the tidal flow with a pronounced flood-ebbchannel system; and (2) interaction of the tidal flow with an ir-regular bathymetry. Residual ebb-flood channel circulation is an

nt and Institutions, UNESCO-ox 3015, 2611 AX Delft, The

uyen).

All rights reserved.

important large-scale mixing mechanism for moving pollutantsand transporting salinity upstream against a mean outflow offreshwater as shown in the Western Scheldt (Savenije, 2005, pp.116–117). The Western Scheldt (see Fig. 1), the Netherlands, is anestuary with a distinct ebb-flood channel system. Similar types ofestuaries are found in the Chesapeake Bay, Maryland, USA (Ahnert,1960); the Columbia river estuary in USA; the Pungue estuary inMozambique as well as several small estuaries in the U.K. van Veenet al. (2005) described the Western Scheldt as an estuary witha regular system of ebb and flood channels. Jeuken (2000, p. 24)identified two basic channel types in the Western Scheldt estuary:(1) main channels and (2) connecting channels. The main channelstransport most of the tidal discharge, sediments and salinity. Theyare the largest channels in the system and they appear in twoforms: a curved main ebb channel and a straight main floodchannel. The connecting channels are the smaller channels thateither link two large main channels or a large main channel witha small main channel. The transport function of the connectingchannels is limited.

Fig. 1. The Western Scheldt estuary in the Netherlands, showing six ebb-flood channelloops in ovals.

A.D. Nguyen et al. / Estuarine, Coastal and Shelf Science 79 (2008) 7–168

Savenije (1993a) found that gravitational (density-driven) cir-culation is the main mixing mechanism if salinity gradients arelarge (as occurs in the central part of estuaries). The river flow playsan important role to drive the gravitational circulation, which isaccompanied by both vertical and lateral salinity gradients. In wideestuaries, lateral stratification generally makes the largest contri-bution to density-driven mixing (Fischer et al., 1979). However, inthe seaward part of estuaries, where the density difference is small,the interaction between tide and geometry, which generates re-sidual (horizontal) circulation, is the main mixing mechanism.McCarthy (1993) presented an analysis on residual circulationgenerated by the combined effect of tide and gravitational circu-lation in an estuary with exponentially varying width. Although hisstudy did not consider the important transport mechanism by ebb-flood channel interaction, it clearly demonstrated that tide-drivenmixing is dominant in the downstream part of estuaries, whiledensity-driven mixing is dominant in the upstream part of estu-aries. Density-driven mixing is a function of the salinity gradient,whereas tide-driven mixing is a function of the salinity and thewidth.

Based on comprehensive reviews of Fischer et al. (1979), Zim-merman (1986) and Geyer and Signell (1992), it can be seen thatmany authors have tried different methods to quantify tidal-drivendispersion. Several authors used a decomposition method (e.g.Fischer, 1976; Lewis, 1979; Uncles and Jordan, 1979; Uncles et al.,1985; Pino et al., 1994; Dyer, 1997; or Sylaios and Boxall, 1998). Anumber of authors proposed to determine the effective longitudi-nal tidal-driven dispersion as a function of mixing length and ve-locity. Examples of this approach can be found in Arons andStommel (1951) and Zimmerman (1976). Arons and Stommel pro-posed D¼ kU0l0 (where D, k, U0, l0 are the dispersion coefficient,a constant, velocity scale and mixing length scale, respectively). k isunderstood to be a constant for one estuary. However, because thevalues of k widely varied, it resulted in a very large range of com-puted values. Zimmerman (1976) developed the ‘‘tidal randomwalk’’ theory and model, which considers the Lagrangian motionsin estuaries resulting from the purely advective effects of tidal andresidual currents and takes into account pronounced horizontalresidual circulations generated by tide-topography interactions.Due to the presence of residual eddies, the velocity field cannot beconsidered uniform and in fact it varies considerably over distancesof the order of the tidal excursion. Zimmerman (1976, 1981) derivedan equation for the longitudinal dispersion coefficient, which is

formally the same as the equation of Arons and Stommel. The maininnovation in the equation of Zimmerman is that k has a well-de-fined physical meaning, being a function of l and n (i.e. the di-mensionless parameters reflecting mixing length and tidalvelocity). However, to obtain reasonable values for k, good qualityand detailed observations are needed. Zimmerman applied histheory to a lagoon system (i.e. Dutch Wadden Sea), which does nothave well-defined channels such as occur in alluvial estuariesstudied herein. The fact that the topography of alluvial estuariesobeys geometrical laws creates an opportunity to expand thistheory and make it more predictive.

Later, de Swart et al. (1997) applied the ‘‘tidal random walk’’model to the Ems estuary and obtained good results, at leastqualitatively, in estimation of the dispersion coefficients. However,they recommended that although the simple random walk modelcan be used to get upper bounds for the dispersion coefficients,simulations using a detailed 3D-numerical model are necessary inorder to obtain lower bounds for the values of dispersion co-efficients. This agrees with assessments in Geyer and Signell (1992),which stated that numerical models provide a means of isolatingthe influence of tidal advection from these other processes and theycan simulate the nonlinear effects that produce residualcirculations.

The aims of this paper are to investigate the residual circulationcaused by the interaction of the tidal flow with the flood-ebbchannel system in the Western Scheldt and to study the effect oftidal pumping caused by the flood-ebb channel residual circulationon the salinity distribution. This paper provides a new conceptualmodel and a new analytical equation for this type of tidal-drivenmixing. Firstly, we use a fully three-dimensional hydrodynamicmodel as a ‘‘virtual laboratory’’ and employ a decompositionmethod to characterise the residual ebb-flood channel circulationin the Western Scheldt. Secondly, a conceptual model and an ana-lytical equation determining the dispersion coefficient are de-veloped, which take into account several important parameters ofthe tidal pumping mechanism, such as the tidal pumping efficiency,the tidal excursion and the length of the branches. Subsequently,the newly developed equation is compared to the results of the‘‘virtual laboratory’’. Finally, the equation is applied to observationsin the Western Scheldt. The application yields good results in de-termination of the longitudinal dispersion compared to dispersionvalues obtained from the salt budget and hence it is demonstratedthat the newly developed equation can be applied to a real estuary.

2. Methodology

2.1. ‘‘Virtual laboratory’’ residual salt transport

In order to analyze the characteristics of the residual ebb-floodchannel circulation, a considerable set of data on residual flowcomponents is required. The collection of such information wouldrequire a major operation, consisting of a dense network of moni-toring points over a considerable period of time. Moreover, sub-stantial errors on measurements may result (Lane et al., 1997). Inpractice, it is not feasible to obtain these data from the field.Therefore, in this paper, a fully three-dimensional hydrodynamicmodel (DELFT3D) has been used as a ‘‘virtual laboratory’’ of theWestern Scheldt estuary to provide the data on hydrodynamics andsalinity. For details on the operation of the hydrodynamic modeland the key features of the formulations used, reference is made toLesser et al. (2004).

Some relevant information on the schematisation of the hy-drodynamic model is summarized as follows:

(1) Configuration. In total the grid includes approximately 12,500active points, grid cells sizes vary from 800 m� 800 m at the

1

10

100

1,000

10,000

100,000

1,000,000

0 10 20 30 40 50 60 70 80 90Distance from the mouth (km)

Loop 1 Loop 2 Loop 3

Loop 4

Loop 5

Loop 6

1

10

100

1,000

10,000

100,000

0 10 20 30 40 50 60 70 80 90 100 110Distance from the mouth (km)

Area (m2) Width (m) Depth (m)

Area (m2) Width (m) Depth (m)

Loop 1 Loop 2

Loop 3

Loop 4

Loop 5

1

10

100

1,000

10,000

100,000

0 10 20 30 40 50 60Distance from the mouth (km)

Loop 1

Loop 2

Loop 3

Width (m) Computed Area (m2) Computed Depth (m)

a

b

c

Fig. 2. Shape of considered estuaries, showing the locations of ebb-flood channelloops: (a) Western Scheldt, The Netherlands (b) Columbia, USA and (c) Pungue,Mozambique.

A.D. Nguyen et al. / Estuarine, Coastal and Shelf Science 79 (2008) 7–16 9

open sea to 150 m� 50 m within the estuary and the numberof vertical layers is 10. The bathymetry data used are of 1998and the simulation period is three months.

(2) Boundary conditions. At the downstream tidal boundary weuse a tidal amplitude of 1.75 m (i.e. tidal range of 3.5 m, whichis a typical value for the Western Scheldt estuary), using thegradient boundary conditions (a so-called Neumann boundarycondition, Roelvink and Walstra, 2004) with the tidal period of12.25 h (i.e. semi-diurnal tide). Because the upstream bound-ary conditions are still located in the tidal zone, we imposeda harmonic function for the discharge in order to have a re-alistic discharge profile. This harmonic function is based ona 1D model and has an amplitude of 4200 m3/s and an averagedischarge of 120 m3/s, which is a typical value for the studiedestuary.

The hydrodynamic model is derived from an existing hydrody-namic model, denoted as KUSTZUID, which was set-up, calibratedand verified by Rijkswaterstaat (Dutch Ministry of Transport andPublic Works). As the original KUSTZUID model was already thor-oughly calibrated by Rijkswaterstaat no extensive calibrationappeared to be necessary. The hydrodynamic regime (i.e. waterlevel, velocity and discharge) of the model was calibrated andvalidated on the basis of data in 2000 and 1978, respectively. Thesalinity simulation was calibrated based on available data in theperiod of September 2002. Generally, computed water levels, dis-charges and salinities are in good agreement with observations,while velocity measurements taken in the center of a channel arereasonably well reproduced by the model.

In order to obtain values for the residual salt transport in the‘‘virtual laboratory’’, the decomposition method of Uncles et al.(1985) and Pino et al. (1994) has been employed. A brief summaryon the method is presented in Appendix A. The residual transport ofsalt in the ‘‘virtual laboratory’’ has been computed for separate ebband flood channels as well as for the entire estuarine channel. Asindicated in van Veen et al. (2005) and Jeuken (2000, pp. 31–36),the flood channel is wide and shallow, and the ebb channel isnarrow and deep. The separation of the ebb and flood channels istherefore based on the highest point of the shallow area betweentwo channels. This separation point can be recognized from thetopographical map. The residual salt transport contains three maincomponents: the total residual transport, tidal pumping transportand gravitational transport. For reasons of simplicity, in theremaining sections, the residual salt transport and its componentscomputed by applying the decomposition method will be called the‘‘virtual laboratory’’ results.

2.2. Conceptual model and analytical equation for dispersion byresidual ebb-flood channel circulation

In this section, we present the development of a conceptualmodel and an analytical equation for determining the dispersioncoefficient in estuaries with a distinct ebb-flood channel system.

2.2.1. Analytical relation between loop length and width inebb-flood channel estuaries

We consider three estuaries, which have a typical flood-ebbchannel system. They are: the Western Scheldt in the Netherlands,the Columbia in USA and the Pungue in Mozambique. We defineone ebb-flood channel loop based on its topographical character-istics. A loop is determined by a reach between two points (shallowareas) where the flood and ebb channels meet (see Fig. 1 for thecase of the Western Scheldt). This definition is similar to the‘‘morphological cells’’ defined by Winterwerp et al. (2001) for theWestern Scheldt. The loop length can be considered as the ‘‘effec-tive’’ horizontal wavelength, which is analogous to the definition of

Zimmerman (1976) and it is an important parameter for de-termining residual dispersion (Zimmerman, 1981).

The shape of these three estuaries corresponds very well withthe exponential functions that follow from the concept of idealestuaries (see Fig. 2 and Table 1).

hAi[ hA0i exp�

Lxa

�(1)

hBi[ hB0i exp�

Lxb

�(2)

Table 1Shape of the studied estuaries

Name A0

(m2)B0

(m)h(m)

a(km)

b(km)

Western Scheldt 100,000 10,000 10 28 28Pungue 28,000 6500 4.3 20 20Columbia 40,000 7000 5.7 45 45

A.D. Nguyen et al. / Estuarine, Coastal and Shelf Science 79 (2008) 7–1610

and h¼ h0¼ const.; where CAD (L2) and CBD (L) are the tidally aver-aged cross-sectional area and width at location x from the estuarymouth, respectively. CA0D, CB0D and h0 are the area, width and depthat the mouth. Finally, a (L) and b (L) are the area and width con-vergence length. The cross-sectional area and width are obtainedfrom observations at mean tide. The convergence length, which isthe length scale of the exponential function, is obtained by fittingEqs. (1) and (2) against observations, employing a least square re-gression analysis. For reasons of simplicity, in the following sec-tions, we shall drop the brackets, which denote the tidal average ofa variable, and we shall use A, B, A0 and B0 to represent tidal averagevalues.

It appears that there is a certain width at which separation be-tween ebb and flood channels no longer occurs. For the three ex-amined estuaries (i.e. Western Scheldt, Pungue and Columbia) thiswidth is in the order of 1.5 km (see Fig. 2). The topographical datasuggest that for an ebb-flood channel system the loop length (i.e.the average length of the flood channel and ebb channel in theconsidered loop) and the loop width (i.e. the average width of theconsidered loop) are related. From the available data of these threeestuaries, it is possible to derive an empirical relation between theloop length and loop width. Fig. 3 shows the parity diagram for thefollowing relation:

Lef

b[ aL

�1L

BL

B0LBL

�B0

BL1

��(3)

where Lef (L) is the loop length, aL (�) is a scaling coefficient that isapproximately equal to 0.5. B0 (L) is the width at the estuary’smouth. BL (L) is the channel width at the point where there is nomore ebb-flood channel separation and B (L) is the averaged loopwidth.

We can see in Fig. 3 that Eq. (3) provides a reasonable fit to theobservations in the three estuaries, especially for the WesternScheldt and the Pungue. This equation provides a predictive valuefor the loop length, which is an important parameter for

0.00

0.20

0.40

0.60

0.80

1.00

0 0.2 0.4 0.6 0.8 1L

ef/b (Observed)

Lef/b

(co

mp

uted

)

Scheldt (Netherlands) Columbia (USA)Pungue (Mozambique) Line of perfect agreement

Fig. 3. Parity diagram for loop lengths in ebb-flood systems of three estuaries:Western Scheldt, Columbia and Pungue.

determining the dispersion coefficient. Eq. (3) has been derivedfrom the salinity intrusion equation under the assumption that ifthe salinity gradient is small (in the downstream part of the estu-ary), all dispersion is caused by residual circulation (see Savenije,2005, p. 120).

2.2.2. Conceptual model and analytical equation for the dispersioncoefficient

Savenije (2005, p. 119) used a simplified numerical model toanalyze effects of residual circulation on salinity distribution in anideal ebb-flood channel estuary and to investigate the relationbetween the longitudinal salt transport and relevant parametersand it was concluded that there is a direct relation between thelongitudinal salt transport and the ebb-flood loop length, as well asthe tidal pumping efficiency. Therefore, the loop length and thetidal pumping efficiency are likely the two most important pa-rameters for the longitudinal salt transport.

Based on these findings, a conceptual model has been made tocompute the longitudinal dispersion caused by the large-scale re-sidual circulation in an ebb-flood channel estuary. Considering onesingle loop and looking at the flood channel cross section over oneentire tidal cycle (see Fig. 4), the salt flux Ff (L3T�1) that is conveyedto an upstream cell equals

Ff [ S0AfET

ep (4)

where S0 (�) is the salinity at the beginning of the loop, Af (L2) is thecross-sectional area of the flood channel, E (L) is the tidal excursion,which can be measured as the distance between the HWS and LWSsalinity curves (Nguyen et al., 2007). T (T) is the tidal period. ep (�)is the tidal pumping efficiency (Savenije, 2005, p. 118), which isdefined as

ep [Dv~vf

(5)

where Dv (LT�1) is the difference between the average flood andebb flow velocities in the flood channel. ~vf (LT�1) is the averagevelocity in the entire channel during the flood tide.

Assuming that the pumping efficiency in the flood and ebbchannel is the same, which is justified on the basis of the ‘‘virtuallaboratory’’ results, we can determine the salt flux Fe (L3T�1) indownstream direction through the ebb channel as

Fe [ LS1AeDve [ LS1AeET

ep (6)

where Ae (L2) is the cross-sectional area of the ebb channel andS1 (�) is the salinity at the end of the loop.

In order to close the salt balance, under a steady state situation,the sum of salt fluxes Ff and Fe should equal the salt flux caused byfreshwater discharge.

S0

S1

epS1A

eE

Fe

Ff +Q

f S1

T

= =

epS0 Af

E

Ff

T

=

Ebb channel

Flood channel

Lef

Fig. 4. Conceptual model for ebb-flood channel dispersion.

-6000

-4000

-2000

0

2000

4000

6000

0 10 20 30 40 50 60 70Distance from the mouth (km)

Salt tran

sp

ort

(10

3 to

n/tid

al cycle)

Ebb channel transport Flood channel transportResidual ebb-flood channel transport

Fig. 5. Residual salt transport computed by the virtual laboratory.

A.D. Nguyen et al. / Estuarine, Coastal and Shelf Science 79 (2008) 7–16 11

LS1AeET

epDS0AfET

ep [ LQf S1 (7)

where Qf (L3T�1) is the freshwater discharge, which has a negativequantity.

Assuming that the cross-sectional areas in the ebb and floodchannel are equal, which is justified on the basis of the measuredtopographical data in the Western Scheldt, then Eq. (7) becomes

AfET

epðS1LS0Þ[ Qf S1

or

AfET

epLefDSDx

[ Qf S (8)

If the salinity gradient dS/dx may be considered constant over theloop, Eq. (8) corresponds with the steady state salt transportequation (see Ippen and Harleman, 1961 or Savenije, 1986):

ADdSdx

[ Qf S (9)

Combining Eqs. (8) and (9), we obtain the expression for theeffective longitudinal dispersion resulting from ebb-flood channelinteraction:

Def [Af

AET

epLef (10)

where Def (L2T�1) is the effective longitudinal tidal average dis-persion coefficient caused by the ebb-flood channel interaction.

Eq. (10) is formally analogous to the equation introduced byArons and Stommel (1951) and Zimmerman (1976, 1981), (i.e.D¼ kU0l0), where kfðAf=AÞep, U0fE=T and l0fLef . In the newlydeveloped equation, the tidal-driven mixing is determined by thepumping efficiency, the tidal excursion and the loop length. Theshape of the estuarine geometry (i.e. the loop length and width)plays a significant role in determining the ebb-flood channel re-sidual transport dispersion and this agrees well with the findings ofZimmerman (1981), who concluded that the shape of the coastalgeometry or the bottom morphology is important to any tidal re-sidual flow mechanism.

Combination of Eqs. (10) and (3) yields

Def [Af

AET

epaL

�1L

BL

B0LBL

�B0

BL1

��b (11)

Assuming an exponential shape of the estuary, as in Eq. (2) weobtain

Def [ 4pEbT

1LBL

B0exp

�xb

�1LBL

B0

!(12)

where 4p ¼ aLðAf=AÞep is the scaling coefficient, which depends onthe pumping efficiency of the estuary.

3. Results

3.1. Residual circulation characteristics of the Western Scheldtobtained from the ‘‘virtual laboratory’’ and the conceptual model

Basing on the ‘‘virtual laboratory’’ output, applying the methodof Uncles et al. (1985), we are able to compute the residual salttransport for the entire estuary. The computed results are pre-sented in Fig. 5. One can see that over one tidal cycle, there is indeedan upstream residual salt transport in the flood channels anda downstream transport in the ebb channels. This confirms theassumption underlying in the conceptual model.

If the estuary is in a steady state, the net salt transport over thecross section equals zero. As we can see in Fig. 5, the mean value of

the net salt transport over the cross section is close to zero. Near theestuary mouth, there is a large deviation. This may be due to the seaboundary condition, irregular topographies or non-perpendicularflows close to the estuary mouth. We can see in Fig. 1 that especiallyin the first two loops, there are connecting channels and higherorder meanders, which may cause this strong fluctuations. Giventhe large momentary fluxes in the ebb and flood branches, theresidual flux is expected to have a high relative error.

3.2. Comparison between the conceptual model, the salt intrusionmodel and the virtual laboratory in computing the dispersioncoefficient

We use three methods to compute the dispersion value: (1) theconceptual model and its analytical equation (Eq. (12)); (2) the‘‘virtual laboratory’’ based on the three-dimensional hydrodynamicmodel; and (3) the salt intrusion model of Savenije. For details onthe salt intrusion model, reference is made to Savenije (1986, 1989,1993b) and Nguyen and Savenije (2006). A comparison betweenthese models when applied to determine the tidally averagedlongitudinal dispersion coefficient is presented in Fig. 6. The tidalpumping efficiency coefficient and other relevant parameters usedin Eq. (12) have been obtained from the simulation of the hydro-dynamic model.

The dispersion coefficient computed by the salt intrusion modelreflects the ‘‘total’’ effective dispersion, i.e. the sum of horizontalcirculation and gravitational circulation (indicated by D(Tot) inFig. 6a). The dispersion computed by the conceptual box modelonly refers to the dispersion by the horizontal ebb-flood channelcirculation Def (indicated by D(Pump) in Fig. 6b). The ‘‘virtual lab-oratory’’ dispersion for tidal pumping and gravitational circulationis indicated by triangles (D(PumpVL)) and circles (D(GravVL)) inFig. 6b, respectively.

In the second loop, between 10 and 25 km from the mouth, theDef curve does not agree with the values computed by the virtuallaboratory (i.e. D(PumpVL) in Fig. 6b). However, in the other rea-ches, the agreement is good. After subtracting Def from the totaleffective dispersion, we obtain a curve for gravitational circulationD(Grav), which is presented by the dashed line in Fig. 6b. One in-teresting thing is that, although there is considerable scatter,D(Grav) reasonably fits the gravitational dispersion values com-puted by the virtual laboratory. Looking at Fig. 6b, the dispersionvalues computed by the virtual laboratory indicate that at thedownstream (seaward) reach of the estuary, the gravitational cir-culation plays a minor role in producing dispersion. As expected,

1

10

100

1,000

0 10 20 30 40 50 60 70 80 90Distance from the mouth (km)

TA

D

isp

ersio

n

co

efficien

t (m

2/s)

0

100

200

300

400

0 10 20 30 40 50 60 70 80 90Distance from the mouth (km)

TA

D

isp

ersio

n

co

efficien

t (m

2/s)

D(Pump) D(Grav)D(PumpVL) D(GravVL)D(Tot) D(TotVL)a b

Fig. 6. Salinity and dispersion in the Western Scheldt: (a) log-scale effective tidal average dispersion as function of x. Comparison between salt intrusion model and ‘‘virtuallaboratory’’ results; (b) tidal average dispersion due to tidal pumping and gravitational circulation on linear scale. Legend: D(Tot): dispersion computed by salt intrusion model;D(Pump): dispersion caused by ebb-flood channel circulation (Eq. (12)); D(Grav): dispersion caused by gravitation circulation, D(Grav)¼D(Tot)�D(Pump); D(PumpVL): virtuallaboratory tidal pumping dispersion; D(GravVL): virtual laboratory gravitational circulation dispersion; and D(TotVL): virtual laboratory total dispersion,D(TotVL)¼D(PumpVL)þ D(GravVL).

A.D. Nguyen et al. / Estuarine, Coastal and Shelf Science 79 (2008) 7–1612

Fig. 6b confirms that dispersion by the residual ebb-flood channelcirculation is dominant in the seaward part of a well-mixed ebb-flood system estuary. However, starting from 60 km from the sea(middle of the estuary), it appears that not only the residual cir-culation but also gravitational circulation becomes an importantmixing mechanism. Upstream from the 65 km mark, where theseparation of ebb and flood channels no longer exists, dispersion byresidual ebb-flood channel circulation disappears and dispersionby gravitational circulation is the major mechanism.

Although scattered and fluctuating over the considered reach,the total dispersion (D(TotVL) in Fig. 6a), which is the sum of tidalpumping and gravitational circulation dispersion computed by thevirtual laboratory, agrees with the results of the salt intrusionmodel. This agreement allows us to conclude that despite the dif-ferent approaches applied to the system, the overall result in de-termination of the dispersion is similar. One remark is that whenapplying the method of Uncles et al. (1985) to compute dispersionvalues, the ‘‘virtual laboratory’’ data obtained from the hydrody-namic model give some unexpected small values for the tidalpumping dispersion between 10 and 25 km from the mouth andunexpected high values for the gravitational dispersion at this lo-cation. The relatively high gravitational dispersion and low tidalpumping dispersion may be due to the difficulty of the ‘‘virtuallaboratory’’ to reproduce the salinity field (see Fig. 7a). We shall seein Section 3.3 that the analytical model is better capable to re-produce the observed salinity distribution, and hence the relateddispersion. The discrepancy may also be due to the interactionbetween flood and ebb channels having different salinity or due toeffects of the neglected ‘‘interact’’ processes (Uncles et al., 1985), theSIPS (i.e. Strain-Induced Periodic Stratification) mechanism (Simpsonet al., 1990), tidal trapping (Jay and Musiak, 1994) and effects ofjunction (Abraham et al., 1986).

The conceptual model does not provide a solution to determinethe gravitational circulation. In order to obtain a complete solution,we decide to use the method of Harleman and Thatcher (1974) tocompute the gravitational circulation dispersion. The results areplotted in Fig. 7b (dash line D(GravH&T). We can see that for thevirtual laboratory case of the Western Scheldt, the agreement is notvery good. Particularly, the maximum value of gravitational circu-lation computed by the method of Harleman and Thatcher appearsat the 55-km mark, while the maximum value for this dispersioncomputed by the combination between the conceptual model andthe salt intrusion model lies in the 65-km mark, which is the pointwhere the separation of ebb and flood channels no longer exists.

3.3. Application to observations in the Western Scheldt

We apply the newly developed equation (i.e. Eq. (12)) to ob-servations in the Western Scheldt to validate the expressions forthe dispersion due to residual ebb-flood channel circulation andgravitational circulation under different discharge regimes.

Data of the Western Scheldt have been obtained from Savenije(2005, p. 147), based on observations in October 1975 withQf ¼ 90 m3/s. The salt intrusion curves and the resulting longitu-dinal distributions of the total dispersion, the dispersion by residualcirculation and the dispersion by gravitational circulation are pre-sented in Fig. 7c and d. The ‘‘virtual laboratory’’ case of the WesternScheldt is presented in Fig. 7a and b for reasons of comparison. Itcan be seen that for the observed salinity, the gravitational circu-lation dispersion computed by the method of Harleman andThatcher appears to have the same shape as the dispersion com-puted by the combination of the conceptual model and the saltintrusion model, except that it underestimates the maximum value.

The tidal pumping efficiency values obtained are presented inTable 2. In this table, one can see that for higher discharges, the tidalpumping efficiency is smaller and the salinity intrusion length issmaller as well. It can be explained as follows: Looking at Eq. (5),when the river discharge increases, the mean flood velocity in theflood channel decreases and the mean ebb velocity in the floodchannel increases. As a result, Dv decreases, while ~vf remains moreor less the same. Therefore, the tidal pumping efficiency decreases.A smaller tidal pumping efficiency causes a smaller D0 (i.e. thedispersion coefficient at the mouth). Combined with a higher dis-charge, this gives a smaller a0 (i.e. the mixing coefficient at themouth), and subsequently, a shorter salinity intrusion lengthaccording to the salt intrusion model (Nguyen and Savenije, 2006).This fact is well demonstrated in Table 2.

4. Discussion

It appears that the newly developed equation for computing theebb-flood channel residual circulation dispersion (i.e. Eq. (12))performs well, especially at the downstream section of the estuary.Eq. (12) is dependent on x (i.e. dispersion is a function of x). Thisagrees with findings in Ippen and Harleman (1961) and Savenije(1986, 1989). However, at the limit of the zone where ebb-floodseparation no longer exists, near the point where B¼ BL, theequation does not provide a good solution (see Figs. 6b, 7b and d). Itis understood that after the ebb-flood separation zone, tidal

0

5

10

15

20

25

30

-10 0 10 20 30 40 50 60 70 80 90 100Distance from the mouth (km)

Salin

ity

0

100

200

300

400

0 10 20 30 40 50 60 70 80 90Distance from the mouth (km)

Disp

ersio

n co

efficien

t (m

2/s

)

0

5

10

15

20

25

30

35

-10 0 10 20 30 40 50 60 70 80 90 100 110Distance from the mouth (km)

Salin

ity

0

100

200

300

0 10 20 30 40 50 60 70 80 90 100Distance from the mouth (km)

Disp

ersio

n co

efficien

t (m

2/s

)

TA Salt intrusion model TA Observations

TA Salt intrusion model TA Virtual laboratory

400

D(Tot) D(Pump) D(Grav) D(GravH&T)

D(Tot) D(Pump) D(Grav) D(GravH&T)a b

c d

Fig. 7. Salinity and dispersion computation in the Western Scheldt: (a) salt intrusion computed by virtual laboratory and salt intrusion model corresponding with Qf¼ 120 m3/s; (b)dispersion coefficient and its components related to (a); (c) modelled and observed salinity distribution corresponding with Qf¼ 90 m3/s; (d) dispersion coefficient and itscomponents related to (c). Legend: D(GravH&T): Harleman–Thatcher’s gravitational dispersion, remaining as in Fig. 6.

A.D. Nguyen et al. / Estuarine, Coastal and Shelf Science 79 (2008) 7–16 13

pumping only plays a minor part in producing dispersion and the‘‘virtual laboratory’’ data confirm this. In this zone, where Eq. (12)demonstrates a sudden transition, there should be a smooth tran-sition between tidal pumping dispersion and gravitational disper-sion. Further research needs to be done to improve this relationship.

The performance of the conceptual model and its newly de-veloped Eq. (12) when applied to the ‘‘virtual laboratory’’ and realobservations in the Western Scheldt provides a promising picture.We consider that the Eq. (12) is indeed applicable in the case of theWestern Scheldt, but it may not yet be a good solution for generalapplications. However, the equation is definitely a function of theidentified parameters, such as the tidal pumping efficiency, thetidal excursion and the estuary shape, particularly the width. Theright form of the equation needs to be investigated further whenwe have proper data sets from real world observations.

Table 2Tidal pumping efficiency values for the Western Scheldt case

Parameters Western Scheldt

Year 1987(observations)Qf¼ 90 m3/s

DELFT3D(virtual laboratory)Qf¼ 120 m3/s

Dvv

0.13 0.11

4p 3.1� 10�3 2.6� 10�3

TA salinityintrusion length (km)

105 92

There may be other deficiencies in the approach. Firstly, theconceptual model assumes a constant salt gradient within one loop,while in reality the salt gradient can vary. For the virtual laboratorycase, it appears that �dS/dx gradually increases upstream untilreaching a maximum at 55 km from the estuary mouth, after whichit gradually decreases upstream. For the longest loop of the system(i.e. the 2nd loop with a length of 14.6 km), the ‘‘virtual laboratory’’shows that the �dS/dx value ranges from 1.95�10�4 to 2.3�10�4

between the most upstream and the downstream point of the loop.Secondly, there is a limitation on the length of the tidal excursion inrelation to the loop length. For the first three loops, the loop length islarger than the tidal excursion, therefore it is correct to assume thatwater particles only travel forward or backward within the desig-nated channels (i.e. ebb channel and flood channel). However, if theloop length is shorter than the tidal excursion, then it is not correct toassume that all water particles move within the designated channel.Therefore in this case the conceptual model is less accurate. The‘‘tidal randomwalk’’ theory of Zimmerman could possibly be a betterinterpretation for the mixing mechanism in these short loops.

Throughout this paper, it has been assumed that only physicalprocesses are responsible for the modelled salinity field in thehydrodynamic model, which in turn is used to estimate the saltdispersion. This poses some limitations for the study. Firstly, thenumerical scheme used in the hydrodynamic model has numericaldispersion. However, the numerical dispersion is expected to beminimized due to the use of an Alternating Direction Implicit (ADI)scheme denoted as a ‘‘cyclic method’’ (Stelling and Leendertse,

A.D. Nguyen et al. / Estuarine, Coastal and Shelf Science 79 (2008) 7–1614

1991), which is demonstrated to be computationally efficient inLesser et al. (2004). The numerical dispersion therefore is not sig-nificant in the area of interests (i.e. the estuarine sections with ebb-flood channel system) due to the large effective dispersion. Sec-ondly, as we mentioned in Section 2.1, the computed water levels,discharges and salinities in the model are in good agreement withobservations, while velocity measurements taken in the center ofa channel are reasonably well reproduced by the model. However,the computed velocity results are not in good agreement withobservations on side slopes of channels, where often a transitionoccurs from ebb-directed to flood-directed residual transport. Thisis partly caused by the resolution of the model grid, which is notsufficiently high to resolve relatively steep gradients of the ba-thymetry. In addition, Winterwerp et al. (2006) using a grid cell sizeof approximately 25 m in the area of interest (i.e. at a river bend)indicated that the velocity profiles are very sensitive to the salinitystructure. Therefore as we use a coarser grid size, it may be that thestructure of the flow field is not fully correct and subsequently,there may be an unsatisfactory representation of the salinity field.Thirdly, we do not include the effect of wind in the three-di-mensional model, therefore freshwater may remain too long nearthe downstream boundary without being replaced by ocean water,resulting in a lower salinity near the downstream boundary (seeFig. 7a). Finally, the model does not explicitly include other pro-cesses producing residual currents, such as ‘‘tidal random walk’’ ofZimmerman (1976), or SIPS, which is a mechanism resulting inparticle trapping because of the occurrence of tidal velocityasymmetry and its interaction with the time-varying concentrationfield (Simpson et al., 1990), or nonlinear effects (Tee, 1978). How-ever, the model computes the entire velocity field and in principlesimplicitly takes account of all these processes. Whether this is fullyin agreement with reality may be questioned but the model hasbeen calibrated to correctly reproduce the velocity field, includingresidual circulation.

The newly developed equation does not account for gravita-tional circulation. The method of Harleman and Thatcher providesa reasonable estimate for this type of dispersion. However, it isnoted that the method of Harleman and Thatcher takes into ac-count mostly vertically gravitational dispersion and this is thereason why we only obtain a reasonably good correspondence withthe subtraction of the total dispersion and the ebb-flood channeldispersion (see Fig. 7b and d). Nevertheless, it is advised to considerthe newly developed equation together with the method of Har-leman and Thatcher, in order to obtain the full dispersion in an ebb-flood channel estuary. It is also suggested that the newly developedequation can be applied to other ebb-flood system estuaries like thePungue in Mozambique and the Columbia estuary in U.S.

The dispersion values computed from the virtual laboratoryappear to agree fairly well with the other two models (i.e. theconceptual model and the salt intrusion model). When computingthe dispersion coefficient on the basis of the ‘‘virtual laboratory’’salinity distribution, the values obtained depend on the salinitygradient. This gradient is very sensitive to errors in the salinities atsubsequent grid points. As a result, we notice that a small variationin the salinity values along the estuary can lead to large fluctuationsin the computed dispersion values, as one can see in Fig. 6a and b.The average values, however, follow the main pattern. Furthermore,it is possible that the relative error that we make when subtractingfluxes can be very large, particularly if the residual fluxes are smallcompared to the momentary fluxes.

5. Conclusion

This paper presents a study on tidal pumping caused by thelarge-scale residual circulation through ebb and flood channels. Ithas been demonstrated that in the seaward part of strongly

converging estuaries, where the density gradient is small, this kindof tidal pumping is the main mixing mechanism.

Based on a conceptual model, a new equation for dispersion byebb and flood channel interaction has been derived. This newequation takes into account two important factors of the ebb-floodresidual circulation, i.e. the tidal pumping efficiency and the looplength. The new equation has been confronted with ‘‘virtual labo-ratory’’ data of the Western Scheldt estuary in the Netherlandsgenerated by one three-dimensional hydrodynamic model andwith the salt intrusion model of Savenije (1993b) and Harleman andThatcher (1974).

The good performance of the newly developed equation incomparison with the existing models as well as with observationsin the Western Scheldt indicates that the equation is indeed ap-plicable in practice. The new equation provides an opportunity toevaluate the large-scale mixing mechanism caused by the residualebb-flood channel circulation, which is difficult to analyze fromfield observations. It is also suggested to further study and to applythis equation to other ebb-flood system estuaries.

Appendix A. Computation of residual salinity transport anddispersion in estuaries

In order to compute the residual salt transport in estuaries,a number of methods have been developed and reported in theliterature (Fischer, 1972, 1976; Lewis,1979; Uncles and Jordan, 1979;Uncles et al., 1985; Pino et al., 1994; Dyer, 1997; Sylaios and Boxall,1998). These methods followed the decomposition approach.

In this paper, we adopt the method of Uncles et al. (1985) andPino et al. (1994) to determine the residual salt transport and itscomponents over entire cross sections. The method is brieflysummarized below.

The cross-sectional instantaneous rate of water transport

The instantaneous rate of water transport Q (L3T�1) througha cross section A (L2), is given by

Q [ Au (A.1)

where u (LT�1) is the instantaneous velocity and the overbar de-notes an average over the cross section.

The residual rate of water transport is defined as

hQi[ hAi½V1DV2� (A.2)

where the CD brackets denote the tidal average of a variable.V1 (LT�1) is the cross-sectional averaged Eulerian residual cur-

rent (or non-tidal drift current)

V1 [ hui[ 1T

Z T

0

1A

Z A

0u da dt (A.3)

If we subdivide the cross section into discrete cells, then Eq. (A.3)is approximated by

V1z1T

Xt

k [ 1

0@1

A

Xn

i [ 1

Xmj [ 1

Aijuij

1ADt (A.3a)

where m is the number of columns, n is the number of rows. Dt andt are the sub-timestep and the number of sub-timesteps within onetidal period. T (T) is the tidal period. Aij (L2) and uij (LT�1) are thearea and velocity of the discrete cells.

V2 (LT�1) is the Stokes drift current.

V2 [D

~A~uE.hAi (A.4)

where ~u ¼ u� CuD and ~A ¼ A� CAD.

A.D. Nguyen et al. / Estuarine, Coastal and Shelf Science 79 (2008) 7–16 15

VL¼ [V1þ V2] is the Lagrangian current, which corresponds tothe tidal average velocity of the freshwater discharge.

The cross-sectional instantaneous rate of salt transport

The instantaneous rate of salt transport QS (L3T�1) througha cross section A, is given by

QS [ AuS (A.5)

where S (�) is the salinity. Furthermore, the residual transport rateof salt is in the form

hQSi[ hAi�VS;1DVS;2DVS;3

�DV

�

S (A.6)

where VS,1 (LT�1) is the cross-sectional residual flux of salt due tothe residual transport of water

VS;1 [ hQihSi=hAi (A.7)

where

hSi[ 1T

Z T

0

1A

Z A

0S da dt (A.8)

If we subdivide the cross section into discrete cells, then Eq. (A.8)is approximated by

hSiz1T

Xt

k [ 1

0@1

A

Xn

i [ 1

Xmj [ 1

AijSij

1ADt (A.8a)

where Sij (�) is the salinity of the discrete cells.Combination of Eqs. A.2 and A.7 yields

VS;1 [ ½V1DV2�hSi[ VLhSi (A.9)

VS,2 (LT�1) represents the cross-sectional averaged residual fluxof salt due to the non-zero correlation between ~Q (L3T�1) and ~S (�),which is called the tidal pumping.

VS;2 [D

~Q~SE.hAi (A.10)

where ~Q ¼ Q � CQ D and ~S ¼ S� CSD.VS,3 (LT�1) represents the cross-sectional averaged residual flux

of salt due to the shear between tidal and residual currents. Thismechanism is driven by density gradients and is called gravitationalcirculation.

VS;3 [�AV 0S0

hAi (A.11)

where V0 (LT�1) and S0 (�) are the deviations of velocity and salinityfrom the cross-sectional value, respectively.

By definition, C~Q D ¼ C

~AD ¼ C~uD ¼ C~SD ¼ 0 and V 0 ¼ S0 ¼ 0.

V�

S represents ‘‘interact’’ processes. V�

S is found to be smallcompared to VS,1, VS,2 and VS,3 (Uncles et al., 1985; Pino et al., 1994).Therefore V

�

S can be neglected.

Steady state dispersion

If a steady state exists, then the freshwater induced currentequals the Lagrangian current and Eq. (A.2) becomes

VL [ uf [

DQf

EhAi (A.12)

where uf (LT�1) is the velocity of the freshwater and CQf D (L3T�1) isthe tidally averaged rate of input of the freshwater.

Under steady state conditions, the net rate of transport of saltacross a section will be zero (i.e. CQSD in Eq. (A.6) equals zero). Thecross-sectional averaged flux of salt transported across the section

by the freshwater discharge ðCQf DCSDÞ will be balanced by thelandward dispersion of salt ðCADDðdS=dxÞÞ (Savenije, 1986). Hence

D [uf hSidhSidx

[VS;1

dhSidx

or using Eq. (A.6) with CQSD ¼ 0

D [ LVS;2DVS;3

dhSidx

[ DtpDDDr (A.13)

where D (L2T�1) is the total effective dispersion coefficient. Dtp

(L2T�1) and DDr (L2T�1) are the dispersion coefficients caused by thetidal pumping and density-driven (gravitational) circulation. Thepositive x-axis points upstream, hence both uf and dS/dx arenegative.

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