Dynamics of the Ems Estuary

Physics of coastal systems

Jerker Menninga

0439738

Utrecht University

Institute for Marine and Atmospheric research Utrecht

Lecturer: Prof. dr. H.E. de Swart

1

Abstract

During the last decades anthropogenic influence affected the Ems estuary to a great extent. It

was observed that the tidal amplitude was amplified at the landward boundary, that the

location of the sediment trapping was shifted land inward and in addition that the suspended

sediment concentration strongly increased. Only little is known about the origin of this change

in sediment dynamics and in the recent years a lot of research is done. This report covers two

different models which both aims to find answers to this question. Chernetskey et al. (2010)

use a tidal approach and they try to explain the change of the sediment distribution with help

of tidal dynamics. In contrast Talke et al. (2009) neglect the effects of the tides and use a

subtidal model. Both models concludes that the asymmetric change of the suspended sediment

concentration (SSC) is caused by a SSC – Current feedback of the system. However Talke et

al. (2009) conclude that changes in the eddy viscosity, stress parameter and depth enhance the

asymmetry, while Chernetskey et al. (2010) conclude that the asymmetry is caused by

changes in the interaction between the M2 tide and the externally forces M4 tide.

2

Content

1 Introduction ......................................................................................................................... 3

1.1 Interest of investigation ............................................................................................... 3

1.2 Regional setting ........................................................................................................... 4

2 Hydrodynamics ................................................................................................................... 5

2.1 Chernetsky et al. (2010) ............................................................................................... 5

2.1.1 Model ......................................................................................................................... 5

2.1.2 Results ....................................................................................................................... 7

2.2 Talke et al.(2009) ...................................................................................................... 10

3 Sediment Dynamics .......................................................................................................... 11

3.1 General model ........................................................................................................... 11

3.2 Talke et al.(2009) ...................................................................................................... 13

3.1.2 Solutions ................................................................................................................. 13

3.1.3 Results ..................................................................................................................... 13

3.3 Chernetsky et al. (2010) ............................................................................................. 15

3.2.1 Results ..................................................................................................................... 16

4 Conclusions ....................................................................................................................... 19

References ................................................................................................................................ 20

3

1 Introduction

1.1 Interest of investigation

In the recent history the Ems estuary is subjected to intensive investigation. This is because in

the last 25 years both flow as well as sediment dynamics has changed. For instance, the tidal

range towards the landward barrier has increased and has become even larger than the tidal

range at the entrance. Furthermore the suspended sediment concentration (SSC) has increased

as well as the location of the sediment trapping, the so called estuarine turbidity maximum

(ETM), which is shifted landward. The mechanics that play a role in the sediment trapping

and the change of the sediment dynamics is not yet well understood, but it is suggested that

the intensive deepening during the last decades is one of the most important factors.

Deepening of the navigational route is necessary, because three harbors and a shipyard for

cruise ships are located at the borders of the estuary (figure. 1) (Talke, 2009).

In 2009 Talke et al. published a research where they explained the shift of the ETM with an

extended version of a subtidal model, formulated by Hansen and Rattray (1965). With this

model a number of mechanisms of sediment trapping were identified. However others

suggested that changes in tidal behavior influence the sediment distribution (Chernetsky,

2010). Since the tidal range has changed for the Ems estuary it is interesting to investigate this

hypothesis. This was done in 2010 by a research of Chernetskey et al. (2010)

The aim of this report is to gain insight in which mechanism changed the tidal dynamics of

the Ems estuary during the last decades and which mechanisms may be held responsible for

the change of location of the sediment trapping. In order to so both researches of Talke et al.

(2009) and Chernetskey et al. (2010) will be covered. The second section presents an

overview of the hydrodynamics and especially the tidal dynamics. The third section covers

the sediment dynamics. Chapter two first discusses the subtidal approach and after that the

effect of the tides will be investigated.

fig. 1: Change of the bathymetry of the Ems estuary between 1980

(blue line) and 2005 (red line) (Chernetsky et al.2010).

4

1.2 Regional setting

The Ems estuary is located in the North of the Netherlands on the border with Germany. It

reaches from the barrier island Borkum to the tidal weir at Hebrum (figure 1) and is partially

well mixed. At the marine side, to the West of Emden, the water depth averages between 10

and 20 meters. To the East of Emden, where the estuary gets significantly narrower, a water

depth of about 7 meters is maintained. The river Ems is discharging fresh water with a yearly

average of 100 m3/s and is responsible for 90% of the fresh water input. Tidal flats cover

about 50% of the estuary and 80% of the Dollard subbasin. The dynamics of the system are

forced by a semidiurnal tide, freshwater flow from the river Ems and the wind. (Talke, 2009)

fig. 2: Overview of the Ems estuary (Chernetsky et al. 2010).

5

2 Hydrodynamics

This section describes the hydrodynamic models used in the papers of Chernetsky et al (2010)

and Talke et al. (2009). There is a fundamental difference between the two approaches, since

Chernetsky et al. (2010) take the effects of tides into account, while Talke et al. use a subtidal

flow.

2.1 Chernetsky et al. (2010)

2.1.1 Model

In their research Chernetsky et al. use the width averaged shallow water equations. First they

assume that the width B(x) of the estuary is exponentially converging and can be written as,

. (1)

Here B0 is the width at the estuary entrance (at x = 0) and Lb the converging length. Next the

water depth is determined by the difference between the bed at z = -H(x) and the water

surface at z = δ(x). The undisturbed water level is set at z = 0. Now the width averaged

shallow water equations in the longitudinal direction can be determined. The momentum

equation reads

(2)

The first three terms on the left hand side represent the advective velocity in the longitudinal

direction, the fourth and fifth terms are the barotropic pressure (related to variations of the

free surface) and the baroclinic pressure (related to variations of the density) gradients. The

last term is the friction component and is related to the vertical eddy viscosity coefficient Av,

which represent small none defined mixing and dispersion processes (de Swart, 2010). The

eddy viscosity is parameterized as

where H0 represents

the water depth at the entrance.

Furthermore the continuity equation is a balance between variations of the longitudinal and

vertical velocities and the width averaged longitudinal velocity

. (3)

In general the density is denoted by contributions due to the salinity, suspended sediment

concentration and temperature. For simplicity Chernetsky et al. chose to neglect the

contribution of the suspended sediment concentration. For the Ems estuary this is

questionable, since the estuary is characterized by high sediment concentrations. Furthermore

the salinity is assumed to be vertically well mixed, which is also accompanied by Talke et al.

(2009). The along channel density distribution is modeled ass

6

. (4)

Here β converts salt into density and the term denote a tidal average of the salinity.

As mentioned earlier the water motion is forced by the tide. The tide consists of a semi

diurnal (M2) and its first overtide (M4) constituent. The elevation of the surface due to tidal

forcing is modeled as

(5)

In order to solve the equations Chernetsky et al. use the no stress and the kinamitic conditions

at the free surface.

and

. (6a)

At the bottom the bed is assumed to be impermeable, in other words; no vertical flow is

possible through the bed. Also partial slip condition is assumed near the bed.

(6b)

7

2.1.2 Results

2.1.2.1 Parameter input

Table 1 gives an overview of the used parameters and how they changed after deepening has

taken place. All the parameters are the result of field measurements, except the vertical eddy

viscosity coefficient Av and the stress parameter s. These values are obtained by calibrating

the model to the observed data. From the table it can be seen that the vertical eddy viscosity

has decreased from 0.019 m2s

-1 in 1980 to 0.049 m

2s

-1 in 2005 and the stress parameter has

decreased from 0.098 ms-1

in 1980 to 0.049 ms-1

. Furthermore the amplitudes of the externally

forced semi diurnal tide M2 and its first over tide M4 decreased over the years from 1.43 meter

to 1.35 meter and 0.25 meter to 0.19 meter, respectively.

2.1.2.2 Water Motion; tidal amplitude

The discussion of the water motion starts with the tidal amplitude. Figure 3 present the model

solution for the amplitude and the phase difference between the horizontal and vertical

amplitude of the semidiurnal tide. From figure 3a a clear amplification of the amplitude near

the weir at Hebrum is visible. The amplitude of the tide increases from approximately 1.03

meter in 1980 to 1.52 meter in 2005. The vertical amplitude at the entrance on the other hand

decreases from 1.43 meter to 1.35 meter. Furthermore from figure 3b it can be seen that the

phase difference between horizontal and vertical tidal amplitude tend to 90o at the weir for

both 1980 and 2005. This indicates that at the weir the tidal wave is characterized by a

standing wave (de Swart, 2010). Comparison with the 2005 solution shows that the phase

difference through the whole estuary is shifted towards 90o, so in 2005 the tidal wave has

become more resonant through the whole estuary.

Table 1: Parameter input. (Chernetsky et al. 2010)

8

Chernetskey et al. (2010) investigate the tidal amplification by adopting two different

scenarios. The model input (table 1) shows that the main difference between 1980 and 2005 is

the change in water depth, the change of the vertical eddy viscosity and the change of the

stress parameter.

For the first scenario they fix the eddy viscosity and the stress parameter to the 1980 values

and they use the 2005 bathymetry. The second scenario is the opposite; the eddy viscosity and

the stress parameter are set to 2005 conditions and for the depth they use 1980 values. The

result for the semidiurnal M2 tide is represented in figure 3 as the solid (2005 bathymetry and

1980 Av and s) and dashed ( 2005 Av and s and 1980 bathymetry) black line. Figure 3a shows

that the amplitude is amplified when only deepening is taken into account, however the

amplitude is amplified even stronger when the eddy viscosity and the stress parameter are

reduced. Figure 3b shows that deepening of the system increases the phase difference between

the vertical and horizontal amplitude, hence the system becomes closer to resonance.

Reduction of the eddy viscosity and the stress parameter has the same contribution to the

relative phase shift. From this Chernetsky et al.(2010) concluded that both deepening and the

reduction of the eddy viscosity and the stress parameter contribute to the amplification of the

M2 tide.

The same procedure is carried out for the first overtide M4 (figure. 4). First it is recognized

that also the M4 tide is amplified from 1980 to 2005. Figure 4a shows the amplitude of the

vertical tide. The figure shows that the reduction of Av and s contributes stronger to the

increase of the vertical amplitude than the scenario where only deepening is taken into

account. Figure 4b shows that also for the M4 tide the system becomes closer to resonance. In

contrast to the vertical M4 amplitude, both scenarios contribute to the relative phase shift.

fig. 3: a) Modeled change of the tidal M2 amplitude as function of the estuary length due to the

deepened scenario and the reduction scenario. b) Modeled change of the relative phase of the

M2 tide as function of the estuary length of the deepened scenario.

9

From the sensitivity test can be concluded that, the deepening, the reduction of the eddy

viscosity and the reduction of the stress parameter contribute to the amplification of both the

M2 as the M4 tide. However, the amplitude of the M4 tide is mainly amplified by the reduction

of Av and s. This result is reasonable, since in general tidal waves lose energy due to friction at

the bottom and internally by small internal eddy’s. When the channel is deepened the tidal

wave is less affected by the bottom and the wave loses its energy slower, hence the amplitude

is less damped (de Swart, 2010). Secondly the stress parameter and the eddy viscosity are

reduced. It is believed that this results from dredging of the channel. By dredging sediments

are loosened and more sediment gets suspended. This results in a smooth and thin layer of

suspended sediment above the bed, which decreases the bottom roughness. The increase in

resonance of the system and the associated standing character of the tidal wave, should result

in a decrease of the tidal return flow. This result is also gained from the model and is shown

in figure 5.

fig. 4: a) Modeled change of the tidal M4 amplitude as function of the estuary length due to the deepened

scenario and the reduction scenario. b) Modeled change of the relative phase of the M4 tide as function of the

estuary length of the deepened scenario.

fig. 5: Change of the tidal return flow between 1980 (left) and 2005 (right)

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2.2 Talke et al. (2009)

In their research Talke et al. (2009) developed a model based on the gravitational model of

Hansen and Rattray (1965). The model that is used assumes subtidal flow. This means that the

model is tidally averaged and it is assumed that the salinity (s) is well mixed in the vertical

and that the viscosity Av is constant. Another important assumption is morphodynamic

equilibrium. This means that it is assumed that there is no net import or export of sediments.

In other words, the model assumes a ‘tank’ of available sediment and it calculates how the

sediment is distributed over the area. Furthermore gravitational circulation is generated by a

balance between the baroclinic and barotropic pressure gradients. So the total pressure

gradient has to be equal to zero. The resulting balance is derived from the momentum balance

and hydrostatic balance, which respectively can be written as (de Swart, 2010)

(7)

(8)

The total pressure gradient can be written as

(9)

So the total pressure is written as the sum of the atmospheric pressure gradient, the barotropic

pressure gradient and the baroclinic pressure gradient (de Swart, 2010). Talke et al. (2009)

neglected the Coriolis force in equation (7) and the circulation due to the atmospheric

pressure gradient in equation (9). This result in the following balance

∫

,

- (10)

Furthermore the horizontal along channel velocity is written as

∫

. (11)

So the total width averaged along channel flow through a cross section of width b and depth H

is equal to the freshwater flow.

The density is described as a function of the salinity and suspended sediment concentration.

This is in contrast to the previous model, where the density was only a function of salinity

(12)

Again β is a quantity that converts the salinity into density and γ is the relative density of the

suspended sediment to the water in order to convert the suspended sediment concentration to

density.

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3 Sediment Dynamics

3.1 General model

Before the differences for the sediment dynamics between the tidal and subtidal approach can

be discussed, first an expression for the distribution of suspended sediment has to be derived.

Talke (2009) and Chernetsky (2010) use similar approaches.

The starting point is the concentration equation:

. (13)

This equation is a balance between the time evolution of the suspended sediment

concentration c and the divergence of the sediment flux F. The sediment flux can be

decomposed into three parts: the advective flux, the settling flux and the diffusive flux, these

are given by

(14a)

(14b)

(14c)

Here u represents the horizontal velocity field, w the vertical velocity component, ws the

settling velocity and Kh and Kv are respectively the horizontal and vertical eddy diffusion

coefficients (de Swart, 2010). Using equations (14a) to (15c) in equation (13) results in the

sediment concentration equation

( )

(

)

(

) (15)

The y-dependence is dropped, since Talke et al. (2009) assumes no variation over the width of

the estuary, while Chernetsky et al. (2010) take a width average. Nevertheless both

approaches result in the above equation. It is also realized that there is no flow and no

sediment flux through the top and bottom boundary, which result in the following conditions:

, (15a)

,

- (15b)

,

- (15c)

From these conditions the vertical sediment distribution can be derived. This is done be

vertically integrate the vertical sediment flux

∫ ,

-

(16)

12

This describes a balance between the deposition of sediment and the stirring due to eddies.

The solution for this is (de Swart (2010)

∫

(17)

Equation (17) is still not completely solved, since the bottom concentration Cb is still

unknown. To resolve this problem, equation (15) is integrated over depth, and it is realized

that there is no flow and no sediment flux through the top and bottom boundary, so the

vertical sediment flux vanishes. This result in

∫ ,

-

(18)

B is a constant of integration and is determined by the difference between erosion (E) and

deposition (D), hence B=E-D. However it is assumed that there is no net inflow or outflow of

sediment so B has to be zero. In other words, there is no along channel variation of the

vertically integrated horizontal flux, hence

∫ ,

-

(19)

This is called morphodynamic equilibrium (Talke et al. 2009).

13

3.2 Talke et al. (2009)

3.1.2 Solutions

In order to find an expression for the horizontal along channel velocity, the momentum

equation (eq.10) is integrated twice with respect to z. The resulting equation depend on the

salinity gradient, the bottom turbidity gradient and the surface slope. Finally the equation

reads

{ } (20)

So the along channel circulation depends on circulations due to baroclinic, suspended

sediment and freshwater contributors. This result is an extension on the known gravitational

model, presented by Hansen Rattray (1965), since circulation due to sediment is also taken

into account. When the along channel sediment gradient is set to zero, the original model is

recovered.

Next substituting (20) and the equation for morphodynamic equilibrium (19) into equation

(17) results in the solution for the bottom sediment distribution Cb

*

,

- + (21)

From the described solutions it becomes clear that both the along channel circulation, as well

as the sediment distribution vary in a cubic sense with the water depth. This result implies that

deepening of the channel has a nonlinear effect on the circulation and the sediment

distribution.

3.1.3 Results

To find how the sediment system reacts on changes of the various parameters, a sensitivity

study is done. Figure 6 shows the sensitivity test for the river discharge, vertical mixing and

water depth respectively. Figure 6a shows that when the river discharge increases, the

turbidity maximum shifts downstream. This is because when the river discharge weakens, the

near bottom currents (e.g. the salinity driven currents) that have an upstream direction become

more important and transports sediment in the upstream direction. In contrast, low discharge

values enhance the asymmetry of the sediment distribution. This can be understood by

evaluating equation (20). For low discharge values, the depth-depended terms become

relatively more important. Since the depth has a cubic relation, asymmetry in the sediment

distribution is enhanced. The same pattern can be seen for changing the mixing term (figure.

6b). For large mixing values, the sediment is able to stay longer in suspension. This result in

excluding sediment from the system. Also, decreasing of the mixing term result in a relative

increase of the asymmetric depth. As expected an opposite trend is shown for depth variations

(figure. 6c). Increasing the depth result in stronger asymmetry of the sediment distribution and

an upstream shift of the turbidity maximum.

14

fig. 6:Sensitivity study of a) the freshwater discharge, b) the vertical eddy viscosity and vertical eddy diffusivity and c) the

depth. (Talke et al. 2009)

15

3.3 Chernetsky et al. (2010)

From the previous paragraph it became clear that changing the depth, river discharge and

vertical mixing parameters affected the sediment dynamics in for the subtidal approach. The

Ems estuary however is a tidal embayment, so it is convenient to study the effect of the tide to

the sediment distribution. Chernetsky et al. did this in their research, with help of the same

model described in section 3.1. However, since tides are now allowed in the system, equation

(19) has to be tidally averaged. First it is realized that the that the velocity and the sediment

concentration can be written as the sum of the mean and the perturbation

(22)

(23)

Combining equation (22) and (23) result in

(24)

Next the boundary conditions state that there is no sediment flux into or out of the domain, so

when taking the tidal average of (24) the first term on the right hand side will vanishes. Using

this, Chernetsky et al. (2010) found an expression for the tidally averaged morphodynamic

equilibrium equation

⟨∫ (

)

⟩ (25)

where δ represents the free water surface and ⟨ ⟩, represents a tidal average.

For their analysis, Chernetsky et al. (2010) describe equation (25) in terms of the erosion

coefficient a(x), which is the horizontal distribution of the total erodible material. In order to

do so two new function are introduced, F which represent the vertically integrated horizontal

distribution due to the turbulent eddies and T which represents the vertically integrated

horizontal distribution due to horizontal flow transportation

⟨∫

⟩ (26)

⟨∫ (

(

))

⟩ (27)

Using equation (26) and (27) into equation (25) result in a differential equation for the erosion

coefficient

(28)

The transport parameter T can schematically be represented as the sum of the transport

contributors from the residual flow Tres, the semi diurnal tide TM2 and its first overtide TM4 and

transport due to diffusion Tdiff: T = Tres+ TM2+ TM4+ Tdiff.

16

3.2.1 Results

From equation (28) the location of the estuary turbidity maximum (ETM) can be determined.

Since the available eroded sediment concentration reaches maximum values at the ETM, the

erosion coefficient a will also reach its maximum. In addition the gradient of the erosion

coefficient will be zero at the ETM. Since a reaches its maximum, the transport parameter

should be zero in order to obey morphodynamic equilibrium. So the four contributors of the

transport parameters should balance at the ETM.

In order to investigate the relative importance of the different transport contributors,

Chernetsky et al. plotted the different transport functions for course and fine silt for both 1980

and 2005 (fig. 7). From the figure it can be seen that for fine silt between 1980 and 2005, the

ETM was shifted about 20 kilometers land inward. This shift can almost completely be

attributed to transport due to the M2 tide, since this transport function remains constant stream

upward, almost until 50 kilometers. This is in contrast with the 1980 situation, where this

function reaches its maximum around 10 kilometers and decreases almost linear after that.

For course silt the plot gives a different view. In 1980 the ETM was located around 10

kilometers from the entrance. The M2 contributor was the only agent that transported

sediment stream upward and decreases rapidly behind the ETM. In 2005 however two ETM’s

are visible. The first is located about 15 kilometers from the entrance and the second around

45 kilometers. For the first ETM the M2 component is the main contributor. In agreement with

fig. 7:Dimensionless transport function and its components. a and b show fine silt and c and d

coarse silt. (Chernetskey et al. 2010)

17

1980 it decreases close to the entrance, however in 2005 it reaches a certain positive constant

value at the location of the first ETM. After that it remains constant for several kilometers

before it decreases further and becomes an exporting contributor. Another difference between

1980 and 2005 is the shift of the transportation due to the M4 tide. Until 20 kilometers the M4

contributor becomes less negative and after that it becomes more positive. As a result the M4

tide transports sediment stream upward in the second half of the modeled area. Together with

the M2 contribution, this results in the second ETM.

From this result Chernetsky et al. concluded that the change of the sediment trapping location

is mainly the result of changes in transport due to the M2 tide, both for fine and coarse silt. To

understand which mechanism is responsible for the change of the transportation by the M2

tide, the TM2 component is decomposed in two main components; the interaction between the

M2 velocity and both the M4 and the residual velocity. This is denoted by and

respectively. The relative importance of the two transportation components to the change of

TM2 is shown in figure 8. From the figure it can be seen that both for fine and coarse silt the

component shows no significant changes between 1980 and 2005. So it can be concluded

that this component is not responsible for the change of TM2. The profile for however

remains for both fine and coarse silt at higher values in 2005 than it did in 1980. Furthermore

it follows the profile of TM2 in 2005.

fig. 8:Components of the dimensionless transport function TM2 (Chernetskey et al.

2010)

18

With this result the component is decomposed further in advective contribution, free

surface contribution, no-stress contribution and the M4 external forcing. This can

schematically be represented as:

, respectively. In order to

investigate which of these components contributes to the change of and as a consequence

to the change of TM2, figure 9 is analysed. From the figure it can be seen that for both fine and

coarse silt the external forcing of the M4 interaction with the M2 velocity is the main

contributor to the change of . Where the other contributors show no significant change

between 1980 and 2005, the remains positive longer and as a consequence is responsible

for the change of .

From this Chernetsky et al.(2010) concluded that the change of the sediment trapping is the

main result from changes in the asymmetry of the tides.

fig. 9: Components of the dimensionless transport function 𝑇𝑀 𝑀

19

4 Conclusions

This report aims to find the processes responsible for the change in tide and sediment

dynamics of the Ems estuary. To this end two papers and two different approaches are

discussed; the paper of Chernestky et al. (2010), which makes use of a tidal approach and the

paper of Talke et al. (2009), which uses a subtidal approach. The biggest similarity between

the two is the conclusion that the change in dynamics is because of asymmetry of the basin.

However, since the basic assumptions differ in great extent, also different conclusions are

drawn.

Since tides are not considered in a subtidal approach, the change of the tide dynamics is only

explained with help of the paper of Chernetskey et al (2010).

To explain the tide dynamics Chernetsky et al (2010) adopted two different scenarios. The

first scenario fixes the eddy viscosity and stress parameter to the initial values and uses the

2005 bathymetry. The second scenario is the opposite.

From this, it is concluded that both the semidiurnal tide as well as its first overtide are

changed due to changes of the stress parameter, eddy viscosity and basin depth. This result is

reasonable, since deepening of the channel, result in less influence of the bottom to the tidal

wave. So less energy is lost due to friction. Secondly the tidal wave is amplified, because the

eddy viscosity and the stress parameter are reduced This can be explained because due to

dredging more sediment is in suspension and a thin film of fluid sediment covers the bottom

and reduces the stress parameter.

Both models are able to explain the change in sediment trapping and both models conclude

that asymmetry is the keyword. Talke et el. (2009) use an extended version of a classical

subtidal model, first formulated by Hansen and Rattray. The extension also deals with

sediment circulation. With this model it is concluded that the change of sediment trapping,

result from changes of the freshwater discharge, eddy viscosity and depth.

Chernetsky et al. tried to relate the change in sediment trapping by changes in tidal

characteristics. They modeled the contributions to sediment transport of both the M2 and M4

tides. From this they concluded that the change of sediment trapping was a result of the

change of transport contributions of the M2 tide. After this they subdivided this contribution

further to M2 interactions with the first overtide velocity and the residual velocity. From this it

was concluded that the interaction between the M2 and M4 velocity infects the location of the

sediment trapping. From another subdivision it was concluded that interactions of the

externally forced overtide and the semidiurnal tide are responsible of the change in sediment

trapping.

20

References

Chernetskey, A.S., Schuttelaars, H.M., Talke, S.A., 2010, The effect of tidal asymmetry and

temporal settling lag on sediment trapping in tidal estuaries, Ocean Dynamics 60:1219-1241

Hansen, D.V. , Rattray M. jr., 1965, Gravitational circulation in strait and estuaries. Journal

of Marine Research 23: 104-122

de Swart, H.E., 2010, Physics of Coastal Systems, lecture notes. IMAU, Utrecht University

Talke, S.A., de Swart, H.E., Schuttelaars, H.M.,2009, Feedback between residual circulations

and sediment distribution in highly turbid estuaries: An analytical model. Continental Shelf

Research 29:119-135