# The effect of tidal asymmetry and temporal settling lag on sediment trapping in tidal estuaries

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## Abstract

Over decades and centuries, the mean depth of estuaries changes due to sea-level rise, land subsidence, infilling, and dredging projects. These processes produce changes in relative roughness (friction) and mixing, resulting in fundamental changes in the characteristics of the horizontal (velocity) and vertical tides (sea surface elevation) and the dynamics of sediment trapping. To investigate such changes, a 2DV model is developed. The model equations consist of the width-averaged shallow water equations and a sediment balance equation. Together with the condition of morphodynamic equilibrium, these equations are solved analytically by making a regular expansion of the various physical variables in a small parameter. Using these analytic solutions, we are able to gain insight into the fundamental physical processes resulting in sediment trapping in an estuary by studying various forcings separately. As a case study, we consider the Ems estuary. Between 1980 and 2005, successive deepening of the Ems estuary has significantly altered the tidal and sediment dynamics. The tidal range and the surface sediment concentration has increased and the position of the turbidity zone has shifted into the freshwater zone. The model is used to determine the causes of these historical changes. It is found that the increase of the tidal amplitude toward the end of the embayment is the combined effect of the deepening of the estuary and a 37% and 50% reduction in the vertical eddy viscosity and stress parameter, respectively. The physical mechanism resulting in the trapping of sediment, the number of trapping regions, and their sensitivity to grain size are explained by careful analysis of the various contributions of the residual sediment transport. It is found that sediment is trapped in the estuary by a delicate balance between the *M* _{2} transport and the residual transport for fine sediment (\(\emph{w}_s=0.2\) mm s^{ − 1}) and the residual, *M* _{2} and *M* _{4} transports for coarser sediment (\(\emph{w}_s=2\) mm s^{ − 1}). The upstream movement of the estuarine turbidity maximum into the freshwater zone in 2005 is mainly the result of changes in tidal asymmetry. Moreover, the difference between the sediment distribution for different grain sizes in the same year can be attributed to changes in the temporal settling lag.

## Keywords

Idealized model Ems estuary Estuarine turbidity maximum Tidal asymmetry Temporal settling lag Shallow water equations Sediment dynamics Morphodynamic equilibrium condition## 1 Introduction

In many estuaries, regions are found with sediment concentrations exceeding those directly upstream or downstream. The region where the highest sediment concentration is found is called the estuarine turbidity maximum (ETM). Quite often multiple peaks of high concentration are found in estuaries (see, e.g., Lin and Kuo (2001) and references therein). This trapping of sediment is the result of the complex interaction of the water motion (forced by tides, river discharge, density gradients) and sediment dynamics (availability of sediment, sediment size, flocculation). At the locations with high sediment concentrations, depleted levels of oxygen (and hence degraded environmental conditions) are often observed (De Jonge 1983; Talke et al. 2009a). Since anthropogenic and natural changes (e.g., sea-level rise, land subsidence, etc.) in estuaries can influence the locations where sediment is trapped and/or the amount of sediment being trapped, it is important to understand the physical mechanisms resulting in the trapping of sediment and how these mechanisms are influenced by changes to the system.

Model input parameters representing 1980 and 2005 measurements carried out along the Ems/Dollard estuary, respectively

Parameter | Symbol | Dimension | 1980 | 2005 |
---|---|---|---|---|

Semi-diurnal tidal frequency | | s | 1.4 × 10 | |

Gravitational acceleration | | m s | 9.8 | |

| | psu | 7.6 × 10 | |

Ref. density | | kg m | 1,020 | |

Sediment density | | kg m | 2,650 | |

Length of the estuary | | km | 63.7 | |

Convergence length | | km | 30 | |

Water depth at the entrance | | m | 12.2 | |

| \(A_{M_2}\) | m | 1.43 | 1.35 |

| \(A_{M_4}\) | m | 0.25 | 0.19 |

Relative phase at the entrance | | degrees | − 170.9 | − 174.6 |

Vertical eddy viscosity coefficient | | m | 0.019 | 0.012 |

Stress parameter | | m s | 0.098 | 0.049 |

River discharge | | m | 65 | |

Along-estuary residual salinity gradient | < | psu m | 0.5 × 10 | |

Settling velocity | | m s | 0.0002 − 0.002 | |

Horizontal diffusivity | | m | 100 |

To understand this kind of changes in estuarine dynamics, large-scale numerical models are often applied (see, e.g., Weilbeer 2007 for the Ems estuary). Because 3D numerical models are computationally expensive and include the sum of many processes (e.g., baroclinic circulation, tidal straining, tidal pumping, flocculation, settling and scour lag, nonlinear interactions, etc.), it becomes difficult to isolate the magnitude and importance of particular processes or investigate parameter sensitivity. These limitations motivate the development of idealized models in which specific physical mechanisms can be studied in isolation. The advantage of an idealized model is that it is complex enough to reproduce the physical behavior of the system well, yet it is relatively simple to construct analytical solutions using standard mathematical techniques, avoiding high computational costs.

Various mechanisms have been identified that can result in the trapping of sediment. For example, using a tidally averaged numerical model, Festa and Hansen (1978) investigated the convergence zone of sediment due to the balance between gravitational circulation (Hansen and Rattray 1965; Officer 1976) and freshwater discharge. Recently, the Festa and Hansen approach was extended to include the influence of longitudinal suspended sediment concentration gradients on the tidally averaged flows and the trapping of sediment (Talke et al. 2009b). The importance of tidally varying processes on the formation of residual (tidally averaged) flows and sediment fluxes is discussed in many publications (Simpson et al. 1990; Geyer 1993; Jay and Musiak 1994; Burchard and Baumert 1998). Recently, an idealized model was developed to study the depth-dependent water motion on the tidal time scale in an estuarine cross section (Huijts et al. 2009) and its interaction with the entrapment of sediment in the lateral direction (Huijts et al. 2006). However, no such model has been developed that describes these processes in the longitudinal direction.

The main aim of the present paper is to develop an analytical model that simulates the along-channel flows resulting from various forcings such as the semi-diurnal and first overtide external forcing, the horizontal density gradient, and the river discharge. Based on the water motion, we calculate the sediment concentration in morphodynamic equilibrium in the estuary. Using the model, we are able to investigate the relative importance of various forcing mechanisms and parameters (e.g., tidal dynamics, sediment grain size, residual flows, etc.) on the changes that occurred between 1980 and 2005. In this paper, we address the following research questions. Why did the tidal range increase by 1.5 m in the upstream reaches between 1980 and 2005? Which mechanisms result in the trapping of sediment and in the variation of the trapping location for sediment with a different grain size under the same hydrodynamic conditions? Why is there an upstream shift of the turbidity zone in 2005 compared to 1980?

In Section 2, we discuss the model equations we use to model the water motion and sediment dynamics: the width-averaged shallow water and advection-diffusion equations, respectively. We make a perturbation analysis and provide analytic solutions of the problem in Section 3. Comparison with observations and results is provided in Section 4, followed by a discussion in Section 5 and conclusions in Section 6.

## 2 Model formulation

*x*= 0, and the weir is found at

*x*=

*L*(see Fig. 1). The estuary is assumed to be exponentially converging, i.e., the width

*B*(

*x*) of the estuary is given by

*B*

_{0}the width of the estuary at the seaward side and

*L*

_{ b }the exponential convergence length. The bed profile is described by

*z*= −

*H*(

*x*),

*z*= 0 denotes the undisturbed water level, and

*z*=

*ζ*(

*t, x*) denotes the water surface.

Here, *x*(*u*) and *z*(*w*) denote the along-channel and vertical coordinate (velocity), respectively. Time is denoted by *t*, *g* ∼ 10 m^{2}/s is the gravitational acceleration, \(\rho_0\sim\text{1,020}\) kg m^{ − 3} is the reference density, and *A* _{ v } is the vertical eddy viscosity coefficient.

*ρ*(

*x*) and varies in general due to the salinity

*s*, suspended sediment concentration

*c*, and temperature. To simplify, we neglect the density gradients caused by suspended sediment concentration (SSC) and temperature and assume that the temporal variations of the salinity field are small compared to the time-averaged salinity field. Furthermore, we assume that the salinity is vertically well-mixed. Hence, the along-channel density

*ρ*is modeled as

*β*∼7.6 × 10

^{ − 4}psu

^{ − 1}converts salt to density and the angular brackets < . > denote a tidal average. Here, <

*s*(

*x*) > is the observed along-channel time- and depth-averaged salinity profile which describes the gradual decrease of the salinity from the sea to the river (i.e., the model is diagnostic in salinity). The salinity profile is prescribed as a hyperbolic tangent profile along the estuary that depends on the freshwater discharge (see Talke et al. 2009a).

*A*

_{ v }is parameterized as

*H*

_{0}the water depth at the entrance of the estuary.

*z*=

*ζ*, the boundary conditions are the no stress condition and the kinematic boundary condition:

*z*= −

*H*(

*x*), we assume the bed to be impermeable, i.e.,

*τ*

_{ b }is proportional to friction velocity squared. By linearizing the quadratic friction law (Zimmerman 1992),

*τ*

_{ b }can be related to the velocity at the bed and reads

*τ*

_{ b }=

*ρ*

_{0}

*su*. Hence, the partial slip condition can be rewritten as \(A_\emph{v}u_z=su\) (Schramkowski et al. 2002). Here, the parameter

*s*is the so-called stress or slip parameter that can still depend on the longitudinal coordinate. Following Friedrichs and Hamrick (1996) and Schramkowski et al. (2002), this dependency is taken to be linear in the local water depth, i.e.,

*M*

_{2}) constituent and its first over tide (

*M*

_{4})

*σ*= 1.4 · 10

^{ − 4}s

^{ − 1}is the tidal frequency of the

*M*

_{2}semi-diurnal tidal constituent and \(A_{M_2}\) and \(A_{M_4}\) are the amplitude of the

*M*

_{2}and

*M*

_{4}(externally forced) tidal constituent, respectively. The relative phase

*ϕ*is the phase difference between the

*M*

_{4}and

*M*

_{2}tidal components, defined by \(\phi=\phi_{\zeta_{M_4}}-2\phi_{\zeta_{M_2}}\), where \(\phi_{\zeta_{M_2}}\) (\(\phi_{\zeta_{M_4}}\)) denotes the phase of the

*M*

_{2}(

*M*

_{4}) tidal constituent. Apart from the externally prescribed

*M*

_{4}overtide, overtides are generated internally by nonlinear interactions. The combination of the

*M*

_{2}and

*M*

_{4}constituents results in so-called tidal asymmetry: an estuary is called flood (ebb) dominant if flood currents are stronger (weaker) than ebb currents.

*Q*is prescribed and the tidal discharge is required to vanish

*c*denotes the width-averaged sediment concentration and \(\emph{w}_s\sim 0.2--5\) mm s

^{ − 1}the settling velocity. The turbulent vertical eddy diffusivity coefficient \(K_\emph{v}\) is assumed to be equal to \(A_\emph{v}\). The horizontal diffusivity coefficient is denoted by

*K*

_{ h }. Suspended sediment is transported due to diffusive contributions, temporal (or local) settling lag effects (related to tidal asymmetry and local inertia, see Groen 1967), and spatial settling lag effects (which are related to the finite time for sediment particles to settle, see Postma 1954; De Swart 2009).

*c*

_{ ∗ }is a reference concentration. It is defined as

*ρ*

_{ s }, the dimensionless bed shear stress by

*τ*

_{ b }/(

*ρ*

_{0}

*g*′

*d*

_{ s }) with

*τ*

_{ b }defined in Eq. 7 and the erosion coefficient by

*a*(

*t*,

*x*). Furthermore,

*d*

_{ s }is the grain size of the sediment and

*g*′ =

*g*(

*ρ*

_{ s }−

*ρ*

_{0})/

*ρ*

_{0}is the reduced gravity. The erosion coefficient

*a*(

*x*)∼10

^{ − 5}models the along-channel distribution of easily erodible sediment, available in mud reaches.

*z*= −

*H*(

*x*). The erosion flux is defined in Eq. 11, and the depositional sediment flux is defined by

*E*> − <

*D*> = 0, a condition for morphodynamic equilibrium is obtained by integrating the sediment mass balance equation (Eq. 9) over depth. Using boundary conditions 5, 6, and 10 and averaging the result over a tidal period, the morphodynamic equilibrium condition reads (for details, see electronic supplement S2)

*a*(

*x*). Since the sediment concentration depends linearly on the erosion coefficient, the morphodynamic equilibrium condition 13 can be rewritten as a first-order linear differential equation for the erosion coefficient

*a*(

*x*)

*a*

_{∗}is the average amount of sediment available for resuspension.

## 3 Perturbation analysis and solutions

In this section, we will approximate the system of equations discussed in Section 2 to obtain a (semi-) analytic solution. Here, we will only give a short outline of the procedure used to construct the reduced system of equations, for a detailed description of this procedure, and the solution method used to solve the resulting reduced system of equations, see electronic supplement S3.

First, the relative importance of the various terms in the model equations is established by performing a scaling analysis, using parameter values representative for the Ems estuary (see Table 1, which is discussed in detail in Section 4). One of the dimensionless parameters that appears in the scaled equations is the ratio of the *M* _{2} tidal amplitude \(A_{M_2}\) and the undisturbed water depth *H* _{0}. This ratio, denoted by *ε*, is much smaller than one, i.e., \(\varepsilon=A_{M_2}/H_0\ll 1\). The other dimensionless parameters are compared to this small parameter *ε*.

Next, we approximate the solution of the dimensionless equations and the boundary conditions by expanding the physical variables in power series of the small parameter *ε*. This expansion is substituted in the scaled equations, and terms of the zeroth (first) order of *ε* are collected, resulting in a leading (first-order) system of equations. The leading order system is presented in Section 3.1, the first-order system in Section 3.2, and the morphodynamic equilibrium condition is discussed in Section 3.3.

### 3.1 Leading order system of equations

The first superscript denotes the order of *ε*, and the second superscript is the index of the lunar constituent under consideration, i.e. in leading order, only an *M* _{2} signal is present in the water motion.

*z*= 0, the boundary conditions are given by

*z*= −

*H*(

*x*), the boundary conditions read

The solution of Eqs. 16–19 describes the propagation of a tidal wave in a medium with effectively a homogeneous density and a varying depth. It is solved by reducing the system to a homogeneous ordinary boundary-value problem for the sea surface elevation *ζ* ^{0} (see Ianniello (1977) and the electronic supplement S3.1 for details).

*z*= 0 require no flux through the boundary,

*z*= −

*H*(

*x*), the boundary condition reads

*M*

_{2}tidal signal in leading order, it follows that the concentration has a residual (tidally averaged) component and all constituents with frequencies that are an even multiple of the

*M*

_{2}tidal frequency, hence

*c*

^{0}still depends on the unknown erosion coefficient

*a*(

*x*).

For the solution method and analytical expressions of the semi-diurnal tidal velocity and sediment concentration components, see electronic supplement S3.1.

### 3.2 Higher-order system of equations

In this section, the first-order system of equations is given. The water motion is discussed in Section 3.2.1 and sediment dynamics in Section 3.2.2.

#### 3.2.1 Water motion

*z*= 0, the boundary conditions read

*z*= −

*H*(

*x*)

Careful inspection of Eqs. 24–28 shows that the order *ε* velocity fields \(u^1, \emph{w}^1\) and the sea surface elevation *ζ* ^{1} consist of the residual contributions (\(u^{10},\emph{w}^{10},\zeta^{10}\)) and contributions (\(u^{14},\emph{w}^{14},\zeta^{14}\)) which oscillate with twice the frequency of the semi-diurnal tide. These contributions are discussed separately in the following paragraphs. For the solution method, see the electronic supplement S3.2.

### Residual flow

*u*

^{02}, \(\emph{w}^{02}\),

*ζ*

^{02}were obtained in Section 3.1.

*z*= 0, the boundary conditions are given by

*z*= −

*H*(

*x*), the boundary conditions read

Equation 29 together with boundary conditions 30–32 describe the residual water motion in the estuary which is driven by the residual forcing terms. First, the water motion is forced by the residual constituent of nonlinear interactions of the leading order *M* _{2} tide due to advective contribution (TS). Second, there is a time-independent forcing (GC) due to the presence of a salinity gradient \(\left\langle s\right\rangle_x\) (gravitational circulation). Third, there is the tidal return transport (SD), which is the residual transport that compensates for the correlation between horizontal and vertical water motion (Stokes drift). Fourth, there is the residual constituent due to the no-stress condition at the surface (SC). Finally, there is a river discharge prescribed at the weir, resulting in a residual water motion in the estuary (RI).

*u*

^{10}and

*w*

^{10}and the sea surface elevation

*ζ*

^{10}reads

### First overtide (M_{4}) flow

*M*

_{4}constituent of the water motion is described by the following system of forced equations

*M*

_{4}contribution and the underbraces \(\underbrace{.}_{}\) are the individual

*M*

_{4}forcing terms.

*z*= 0, the boundary conditions are given by

*z*= −

*H*(

*x*), the boundary conditions read

*M*

_{4}tide. These conditions are

System 34 and boundary conditions 35–37b describe the *M* _{4} water motion in the estuary driven by the *M* _{4} constituent of nonlinear interactions of the leading order *M* _{2} tide due to advective contribution (AC), the *M* _{4} transport (FS) that compensates for the correlation between horizontal and vertical water motion, and the *M* _{4} constituent of no-stress condition at the surface (NS). The AC, FS and NS constituents are produced within the estuary and are referred to as the internally generated overtide. Moreover, the water motion is forced by an externally prescribed overtide (forcing term EF). These four components result in a *M* _{4} tidal motion in the estuary.

#### 3.2.2 Sediment dynamics

### 3.3 Morphodynamic equilibrium condition

Here, the first contribution models the residual transport of sediment due to interactions of the residual velocity and the time-averaged sediment concentration. Note that we can decompose this flux even further as we have separate expressions for the residual flow components due to different forcing agents (see Eq. 33). The second term describes the semi-diurnal sediment transport which occurs due to the interactions of the semi-diurnal velocity and the *M* _{2} concentration. The third component represents the first overtide transport of sediment and is a result of the interactions of the first overtide velocity and the *M* _{4} concentration. The horizontal diffusive flux and a flux due to the correlation between the tidal return flow and concentration are represented by the fourth and fifth terms, respectively. Note that, as in the case with the residual sediment flux, we can perform further decomposition of the aforenamed fluxes.

*a*(

*x*), i.e.,

*c*

^{00}=

*a*(

*x*)

*c*

^{00a },

*c*

^{04}=

*a*(

*x*)

*c*

^{04a }, and

*c*

^{12}=

*a*(

*x*)

*c*

^{12a }, where

*c*

^{00a },

*c*

^{04a }, and

*c*

^{12a }are independent of

*a*(

*x*). This results in a linear first-order ordinary differential equation for the erosion coefficient

*a*(

*x*)

*T*, induced by various interactions between the velocity and concentration: \(T_{\text{res}}\) is the transport, which results from the interaction of the residual velocity with the residual concentration; \(T_{M_2}\) is the transport due to the

*M*

_{2}velocity and

*M*

_{2}concentration interaction; \(T_{M_4}\) is the transport due to the

*M*

_{4}velocity and

*M*

_{4}concentration interaction; and \(T_{\text{diff}}\) is the diffusive transport. These contributions are discussed in detail in Section 5.2.1.

## 4 Results

In this section, the model developed in Sections 2 and 3 is used to gain insight into both the hydrodynamics and sediment dynamics in the Ems estuary. There has been a distinct change in both the water motion and sediment dynamics over the past 30 years in this estuary. These changes and the general characteristics of the Ems estuary are briefly discussed in Section 4.1. Using the observations carried out in 1980 and 2005 on the Ems estuary, we will investigate the ability of the model to reproduce the main characteristics of the hydro- and sediment dynamics (Section 4.2), resulting in a physical explanation for why the dynamics changed so drastically. The water motion results are discussed in Section 4.3, followed by Section 4.4 in which the sediment dynamics and the trapping of sediment in the estuary are discussed.

### 4.1 Characteristics of the Ems estuary

*x*is located at Knock, with

*x*increasing toward the weir at Herbrum (see Fig. 2). Moreover, in the remainder of the article, the term entrance denotes the model entrance, which is km 0 in our coordinate system (Knock), and not the geographical entrance of the Ems estuary.

The Ems estuary is exponentially converging with a convergence length *L* _{ b } of 30 km. The system is partially mixed and the landward limit of the salt intrusion varies with the river discharge and is usually found between 20 and 30 km (Talke et al. 2009a).

Approximately 90% of the freshwater input comes from the Ems river, the remaining 10% from other tributaries. The river discharge exhibits seasonal variations. The discharge is approximately 30 m^{3}/s during low flow conditions (June–October) while 150 m^{3}/s is observed during high flow conditions (November–April). The yearly average freshwater discharge is 70 m^{3}/s.

*M*

_{2}tide with an amplitude of 1.43 m. The first overtide had an amplitude of approximately 0.25 m and a relative phase of − 171.9°. The tidal range decreased upstream (near Papenburg it was approximately 2.3 m). Surface measurements registered that average SSC steeply increased from an average of 70 mg/l in Knock (km 0) to a maximum of 400 mg/l in Terborg (approximately km 25) and decreased sharply further upstream (De Jonge 1983).

Between 1980 and 2005, portions of the Ems river were deepened from 4–5 to 7 m. The solid blue line in Fig. 3 depicts the 2005 water depth (WSA Emden). At the entrance, the mean tidal range is now 3.2 m (the spring-neap tidal range is 3.5 and 2.7 m, respectively) and the main tidal constituent is the *M* _{2} tide with an amplitude of 1.35 m. The first overtide has an amplitude of approximately 0.19 m and a relative phase of − 174.6°. The tidal range increases upstream (near Papenburg it is approximately 3.8 m). Measurements between Pogum (km 15) and Rhede (km 59) in 2005–2006 found an average SSC of 950 mg/l with no distinct peak (V. de Jonge, personal communication).

A variety of measurements were used to calibrate the model to present conditions (labeled the “2005” case for convenience) and historical conditions (labeled “1980” for convenience). The variation of *M* _{2} and *M* _{4} water level at eight locations (between km 0, 13, 19.6, 27.4, 37.1, 44.4, 48, and 63.5 in our coordinate system) along the estuary were obtained from measurements over 24 hours by the Wasser and Schiffartsamt (WSA), Emden on February 29th, 1980, during median tidal conditions. Concurrent measurements of water level and velocity from a nearly 14 hour period on April 25, 1984 are used to estimate the historical phasing of flow and pressure. Velocity was estimated using hand held instruments every 15 min at 0.3–0.5 m increments in the water column at three locations on two cross sections located on Ems km 38.5 and 41.1. The phase difference between water level and flow used in the model is obtained from the average harmonic fit of near-surface measurements from the four available digitized data sets and is approximately 65°.

Considerably more data were available to calibrate the “2005” model condition, which is subsequently better constrained. Continuous water level measurements (5–30-min increments) from 2005 and 2006 were obtained from the Niedersaechsicher Landesbetrieb fuer Wasserwirtschaft, Kuesten- und Naturschutz (NLWKN) at the same locations as the 1980 data (between the towns of Knock and Herbrum). Additionally, velocity measurements from surface-moored Aandera RCM-9 probes deployed by WSA Emden were used to define the phase angle between surface tidal flows and water level at a total of six along-channel locations between January–April 2006. Because instruments were serviced occasionally or moved to different along-channel locations, some data gaps exist and the total data set spans approximately 2–2.5 months at each site. Bathymetry for the “2005” and “1980” case were obtained from surveys by WSA Emden in December 2004 and 1984, respectively.

Over most of the modeled estuary, bottom sediment consists primarily of silt (70–80%), with approximately 20–25% fine sand and 5% clay (M. Krebs, personal communication). The channel bed before deepening was primary sandy (Krebs and Weilbeer 2008). Observed flocculation settling velocities in the Ems vary from < 0.0001 to 0.008 m s^{ − 1} (van Leussen and Cornelliese 1993, 1996; van der Lee 2000).

### 4.2 Model setup

Historical bathymetry, digitized from an old charts, was only available between Emden and Herbrum. To be able to simulate the water motion and sediment dynamics for 1980, we have extended the available bathymetry of 1980 from Emden downstream to Knock and from Papenburg up to Herbrum by using the 2005 bathymetry. This is a fair assumption, since channel deepening was mainly carried out between Emden and Papenburg. Moreover, because we focus on flow and sediment pattern on the basin scale, the bathymetries were smoothed using a lowpass filter. Most other model parameters are directly obtained from the observations outlined in Section 4.1 and are summarized in Table 1.

The vertical eddy viscosity coefficient *A* _{v0} and the stress parameter *s* _{0}, which are unknown, are obtained by calibrating the model to the measured data. We minimize the difference between the observed and modeled semi-diurnal tidal amplitude (\(\zeta_{M_2}\)) and the phase difference between the semi-diurnal horizontal and vertical tide (\(\phi_{\zeta_{M_2}}-\phi_{u_{M_2}}\)) in a least square sense. The other observations, such as the residual and *M* _{4} tidal amplitude and velocity at the surface, are then used to validate the model. The resulting vertical eddy viscosity coefficient *A* _{v0} has decreased from 0.019 m^{2} s^{ − 1} in 1980 to 0.012 m^{2} s^{ − 1} in 2005. Apart from this decrease of *A* _{v0}, *A* _{ v }(*x*) (see Eq. 4) decreased everywhere as well compared to the 1980 case. For the stress parameter, we found 0.098 m s^{ − 1} in 1980 and 0.049 m s^{ − 1} in 2005 as best fit.

The river discharge that we use is 65 m^{3}/s. It is assumed that the turbulent vertical eddy diffusivity *K* _{ v } is equal to the vertical mixing coefficient *A* _{ v }. The horizontal diffusivity coefficient *K* _{ h } is taken to be 100 m^{2}/s.

### 4.3 Water motion

*M*

_{2}amplitude increases and the phase difference between the horizontal velocity and water level is closer to 90°.

*M*

_{4}over

*M*

_{2}horizontal velocity at the surface. The solid blue (black) line represents the model results for the 2005 (1980) case, using parameter values given in Table 1 and the red square (cross) marks the observed values for 2005 (1980). The ratio of the

*M*

_{4}over

*M*

_{2}velocity amplitude has decreased from 1980 to 2005 in the first 20 km from the entrance and has increased everywhere else. Figure 5b depicts the relative phase of the sea surface elevation, which is defined as \(\phi_{\zeta_{M_4}}-2\phi_{\zeta_{M_2}}\), where \(\phi_{\zeta_{M_2}}\) (\(\phi_{\zeta_{M_4}}\)) is the phase of the

*M*

_{2}(

*M*

_{4}) tidal elevation. The relative phase defines the duration of the rise and fall of the tide. Both observations and the model suggest that the duration of the falling tide is greater than the rising tide everywhere in the estuary, with no striking quantitative differences observed between 1980 and 2005. In Fig. 5c, the relative phase of the horizontal velocity (\(\phi_{u_{M_4}}-2\phi_{u_{M_2}}\)) is shown. The relative phase between the semi-diurnal tidal velocity and its first overtide, i.e., the

*M*

_{4}velocity, determines whether there is flood or ebb dominance (Aubrey and Speer 1985). If the relative phase is between − 90° and 90°, the estuary is flood dominant; otherwise, the estuary is ebb dominant. Model results presented in Fig. 5c suggest that the ebb/flood dominance changed between the two years. In 1980, there was only a pronounced flood dominance in the last 30 km of the estuary with a small ebb dominated region at the entrance. In 2005, however, the estuary was flood dominated everywhere.

### 4.4 Sediment Dynamics

To reflect the observed variation in the grain size distribution, we modeled fine silt with a settling velocity *w* _{ s } of approximately 0.0005 m s^{ − 1} and a coarser silt with a settling velocity of 0.002 m s^{ − 1}. All other parameters have their default values (see Table 1).

Note that the erosion coefficient *a*(*x*) in the morphodynamic equilibrium condition is determined up to a constant *a* _{∗} (see Eq. 15). Here, we used \(a_\ast = 10^{-5}\) to get the SSC at the surface of approximately 300 mg/l for fine sediment in 1980. An increase (decrease) of the constant *a* _{∗} results in higher (lower) SSC. For both 1980 and 2005, we used the same *a* _{∗} for consistency.

## 5 Discussion

In this section, we will use the analytical model to explain the model results presented in the previous section. Section 5.1 provides an analysis of the hydrodynamic conditions in the estuary. In Section 5.2, we will study the changes of trapping locations between 1980 and 2005 for fixed settling velocity *w* _{ s } and river discharge *Q* and compare the difference in grain sizes for the same year. In Section 5.3, we will discuss the sensitivity of trapping locations to the settling velocity and river discharge.

### 5.1 Hydrodynamics

We start the analysis of the hydrodynamic conditions in the estuary by investigating the difference of the *M* _{2} and *M* _{4} tidal characteristics between 1980 and 2005. As shown in Fig. 4, the dimensionless *M* _{2} tidal amplitude (the ratio of the *M* _{2} tidal amplitude over its value at the entrance) at the landward side increased from approximately 0.74 to 1.12 between 1980 and 2005. Furthermore, the relative phase between the horizontal velocity and water level is closer to 90° in 2005 compared to 1980, i.e., the character of wave has become more that of a standing wave and the estuary is closer to resonance. A similar amplification for the *M* _{4} tide is observed.

We investigate the amplification of the *M* _{2} tide by looking at two different scenarios (details are given in electronic supplement S4). Deepening the estuary to the 2005 depth, but keeping the 1980 *A* _{ v } and *s* parameter values, results in an increase of the dimensionless *M* _{2} tidal amplitude at the landward boundary to 0.89. Keeping the 1980 depth but using the 2005 parameter values for *A* _{ v } and *s* results in the dimensionless *M* _{2} tidal amplitude of approximately 0.99 at the landward side. Hence, we can conclude that, although the reduction of the vertical eddy diffusivity and stress parameter has more effect on the tidal resonance in the case of the Ems estuary, the observed amplification of the dimensionless *M* _{2} tidal amplitude to approximately 1.12 in 2005 is a result of both factors acting simultaneously.

*M*

_{2}velocity phase curves have a similar trend, and they do not change the flood dominance between years (see Fig. 8a). The two contributions to the

*M*

_{4}phase of the horizontal velocity at the surface are shown in Fig. 8b. The blue line denotes the internally generated contribution which is caused by nonlinear interactions, and the black line shows the externally forced contribution. Dashed lines depict the 1980 case and solid lines the 2005 case. The externally generated

*M*

_{4}velocity phase shows qualitatively similar behavior for both years. The character of the internal

*M*

_{4}velocity phase has changed significantly: in the region where we observe ebb dominance in 1980, the phase rapidly increases from approximately − 250° to − 100°, while in 2005 the phase is always about − 100°, resulting in a stronger flood dominance in 2005 compared to 1980. So we can conclude that the main changes in ebb/flood dominance, as observed in the Ems, are a result of changes in the phase of the internally generated overtide.

*M*

_{4}over

*M*

_{2}horizontal velocity at the surface, plotted in Fig. 5a. The

*M*

_{2}and

*M*

_{4}horizontal velocity at the surface are shown in Fig. 9. Figure 9a indicates that the behavior of the

*M*

_{2}velocity component did not change between 1980 and 2005. The only change is an average 12% amplification of the

*M*

_{2}velocity in 2005 compared to 1980. In Fig. 9b, we plotted both the externally forced and internally generated

*M*

_{4}velocity amplitude at the surface for both years. In Section 4.3, we discriminated between two distinct regions. In a region within 20 km of the entrance the first region, located close to the entrance, the

*M*

_{4}/

*M*

_{2}ratio has decreased from 1980 to 2005. This is caused by a local decrease of the

*M*

_{4}internally generated velocity in 1980. Everywhere else we observe the opposite: an increase of the

*M*

_{4}/

*M*

_{2}ratio in 2005 compared to 1980. This can be explained by a rapid damping of the

*M*

_{4}externally forced signal toward the end of the estuary in 1980, which is not the case in 2005. Moreover, the internally generated

*M*

_{4}velocity amplitude is more amplified in 2005 than in 1980 everywhere in the estuary.

*s*(resulting in a wave with a stronger standing wave character) result in a decrease of the tidal return flow. In this case, based on additional model computations, the latter effect is stronger. The lower panels, Fig. 10e, f, represent the river velocity in 1980 and 2005, respectively. The river velocity has decreased by less than 5% in 2005 due to deepening, but there are no significant changes between 1980 and 2005. Therefore, the combination of the increased gravitational circulation and the decrease of the tidal return flow resulted in the residual flow changes described in Section 4.3.

### 5.2 Analysis of the sediment fluxes

In Fig. 7, we observe different sediment trapping regions between different years and/or grain sizes. For fine silt, the trapping region is found at the classical location in 1980, i.e., near the upward limit of salt intrusion. In 2005, the trapping region has shifted further upstream by 19 km. Coarse silt is trapped at the classical location in 1980 as well, whereas in 2005 two trapping regions are observed. One ETM is located at the classical location and the other trapping region is far upstream into the freshwater zone. To understand the physical mechanisms resulting in these different trapping regions, we analyze the sediment fluxes discussed in Section 3.3.

*M*

_{2}concentration with the

*M*

_{2}velocity, the

*M*

_{4}concentration, and the

*M*

_{4}velocity and diffusion (i.e., \(aT_{\text{diff}}+a_xF\)), respectively. Since the system is in morphodynamic equilibrium, these fluxes balance, i.e., the sum of the different contributions vanishes everywhere in the estuary, resulting in no residual sediment transport anywhere. The erosion coefficient that results in this morphodynamic equilibrium (and is calculated using Eq. 42) is the dashed blue line in Fig. 11. The location of the maximum corresponds to the location of the ETM and is at the classical location. In the sediment balance, all fluxes have a significant contribution. There is no approximate balance between the diffusive flux \(a(x)T_{\text{diff}}+a_xF\) and the \(a(x)T_{\text{res}}\) flux as used in (Talke et al. 2009b).

From Eq. 42, it follows that a (local) maximum or minimum in sediment concentration is found in the vicinity of the location where the transport function *T* = 0. At these locations, *da*/*dx* = 0, and since *a*(*x*) itself is nonzero, *T*(*x*) has to be zero. Hence, to determine the ETM locations (i.e., a (local) maximum of sediment concentration) one has to investigate the zeros of *T*. Differences in trapping regions, as observed in Fig. 7, result from an up- or downstream shift of the convergence point or the occurrence of a new location with *T* = 0.

*T*(see Eq. 42). As an example of such a decomposition of the dimensionless transport function

*T*(

*T*divided by its maximum absolute value) and its components for 1980 (fine sediment) are shown in Fig. 12. The blue line represents the total function

*T*and its zero crossing (marked with a blue circle) corresponds to the location of the ETM in Fig. 11. The red, green, black, and magenta lines in Fig. 12 are the contributions to the transport function

*T*, which result from the interaction of the residual velocity with the residual concentration (\(T_{\text{res}}\)),

*M*

_{2}velocity with

*M*

_{2}concentration (\(T_{M_2}\)),

*M*

_{4}velocity with

*M*

_{4}concentration (\(T_{M_4}\)), and diffusion (\(T_{\text{diff}}\)), respectively. From the seaward boundary up to approximately 18 km, the sediment is transported up-estuary mainly by \(T_{M_2}\) (the arrows in Fig. 12 show the direction of the sediment transport, with its color representing the main transport mechanism. Here, the green arrow indicates the import of sediment), while from 18 to 63 km transport is down-estuary mainly by \(T_{\text{res}}\) (the export of sediment is indicated by the red arrow). Hence, there is a convergence of sediment around 18 km. By analyzing the sediment transport components and comparing them for different grain sizes and/or years, we can conclude which fluxes are responsible for the sediment trapping and the observed differences in the sediment trapping between the years (Section 5.2.1) and for different grain size (Section 5.2.2).

#### 5.2.1 Changes between 1980 and 2005

*M*

_{2}transport (\(T_{M_2}\)) and the major export is caused by the residual transport (\(T_{\text{res}}\); see arrows in Fig. 13a, b). For coarse sediment in 1980, the main import of sediment in the upstream region is due to \(T_{M_2}\) (green arrow in Fig. 13c) and the export is due to a combination of \(T_{\text{res}}\), \(T_{M_2}\), and \(T_{M_4}\) transports (

*green/red arrow*). In 2005, coarse sediment was primarily imported in the upstream reaches by \(T_{M_2}\) and exported by \(T_{\text{res}}\), whereas sediment to the downstream ETM is imported due to a combined transport of \(T_{M_2}\) and \(T_{M_4}\) and exported due to \(T_{\text{res}}\) and \(T_{M_2}\) (see arrows in Fig. 13d).

From Fig. 13a, b, we see that for fine silt the global behavior of the transport function *T* changed between 1980 and 2005. Closer inspection shows that this change is mainly due to a change in \(T_{M_2}\). In 1980, \(T_{M_2}\) is approximately constant and positive up until 20 km, after which it starts to decrease. In 2005, \(T_{M_2}\) only starts to decrease after approximately 50 km. Therefore, the transport function *T* becomes negative farther upstream in 2005 than in 1980. This results in a shift of the convergence point between 1980 and 2005 by approximately 19 km.

For coarser silt, we observe one convergence point in the 1980 case and two convergence points in 2005 (see Fig. 13c, d). In both cases, the first convergence point is still at the classical location, even though its position has shifted upstream by approximately 5 km in 2005 compared to 1980. The second convergence point is well into the freshwater region. The change in the transport function *T* is again mainly due to changes in the \(T_{M_2}\) contribution. In 1980, \(T_{M_2}\) becomes negative at approximately km 18, whereas in 2005 this point is located closer to the weir at km 52.

It follows that for both fine and coarse silt the \(T_{M_2}\) contribution play an important role in the changes of the sediment trapping location in the Ems estuary between 1980 and 2005. In the next paragraphs, we analyze the \(T_{M_2}\) contribution in more detail. For detailed analysis of the other components of the transport function, see electronic supplement S5.

To understand which mechanism is responsible for this significant change of \(T_{M_2}\), \(T_{M_2}\) will be decomposed into different components. The *M* _{2} concentration (see Section 3.2.2) is forced by the *M* _{2} component of the bed shear stress. From Eq. 39, it follows that the *M* _{2} component of the bed shear stress is a result of the interaction of both the residual and the *M* _{4} velocities with the *M* _{2} velocity. Contributions of \(T_{M_2}\) that involve the residual velocity components are denoted by \(T_{M_2}^{\text{res}}\) and \(T_{M_2}^{M_4}\) denotes the contribution of \(T_{M_2}\) due to the overtide velocity components (for details, see electronic supplement S5).

*M*

_{4}velocity components that contribute to \(T_{M_2}^{M_4}\) are advective contributions, denoted by \(T_{M_2}^{\text{AC}}\); free surface contribution, denoted by \(T_{M_2}^{\text{FS}}\); no-stress contribution, denoted by \(T_{M_2}^{\text{NS}}\); and the

*M*

_{4}external forcing, denoted by \(T_{M_2}^{\text{EF}}\) (see Eq. 38 and electronic supplement S5 for details). The \(T_{M_2}^{M_4}\) components of the transport function \(T_{M_2}\) are shown in Fig. 15. For both fine silt (Fig. 15a, b) and coarse silt (Fig. 15c, d), we see that the changed behavior of \(T_{M_2}^{M_4}\) (the dashed blue line) is primarily determined by the \(T_{M_2}^{\text{EF}}\) contribution which results from the externally forced

*M*

_{4}tide (the solid magenta line). The other three contributions are much smaller. In 1980, the transport due to the external overtide decreases from km 16, whereas in 2005 an abrupt decrease starts only at approximately km 40. Hence, the main change between 1980 and 2005 is due to the difference in residual sediment transport by tidal asymmetry, resulting in less import of sediment in 1980 compared to 2005, both for fine and coarser silt.

As overall conclusion, it follows that the changes in trapping location between 1980 and 2005, for both years, is a result of changes in the sediment transport due to tidal asymmetry, i.e., changes of the external overtide.

#### 5.2.2 Grain size sensitivity

In the previous section, our focus was on the changes between 1980 and 2005 for sediment with the same grain sizes. Here, we analyze the difference in the sediment distribution for different grain sizes for the same year. As shown by (Groen 1967), the temporal settling lag mechanism is quite efficient in transporting suspended sediment in one direction (upstream in our case) when there is tidal asymmetry.

*T*with (solid lines) and without (dashed lines) the temporal settling lag (TSL) effects in 1980 and 2005, respectively (see Eq. 9). As we can see from Fig. 16a, in 1980, there is virtually no import of sediment into the estuary in the absence of the temporal settling lag (the dashed lines). Temporal settling lag (the solid lines) results in an import of sediment upstream and is, as expected, more efficient in transporting fine silt (from the entrance up to approximately 20 km) than coarse silt (up to approximately 12 km). In the 2005 case shown in Fig. 16b, we observe two trapping locations for both fine and coarse silt in absence of the temporal settling lag mechanism (the dashed lines). One ETM is located at the classical location and the other one further upstream at km 45. For fine silt, the temporal settling lag (the solid lines in Fig. 16b) qualitatively changes the trapping locations: instead of two trapping locations, fine silt is deposited only at one location at approximately km 37. For coarse silt, temporal settling lag is not as efficient and we observe an insignificant shift of the trapping locations upstream (the blue dashed and solid lines in Fig. 16b).

From Fig. 16, we conclude that the difference between the sediment distribution of different grain sizes is a result of the temporal settling lag mechanism.

### 5.3 Parameter sensitivity

*Q*= 65 m

^{3}/s and settling velocity

*w*

_{ s }of 0.0005 and 0.002 m s

^{ − 1}(see Section 5.2.1) gives a good insight into processes in the estuary, but only for these specific parameters. To study the sensitivity of the turbidity maxima to river outflow and settling velocity, we analyze the transport function

*T*for a range of settling velocity

*w*

_{ s }(0.0002–0.002 m s

^{ − 1}) and river discharge

*Q*(20–140 m

^{3}s

^{ − 1}). For each

*w*

_{ s }and

*Q*, we constructed a plot, which is similar to Fig. 12 and determined the location where an ETM occurs. We summarize these locations in Fig. 17a for 1980 and Fig. 17b for 2005. On the

*x*-axis, the location in the estuary is plotted, on the

*y*-axis the settling velocity. If trapping occurs at a certain location for a specific settling velocity, the associated river outflow is given by the color code. If no trapping occurs for a settling velocity at a certain location for a river outflow between 20 and 150 m

^{3}/s, then this (

*x*,

*w*

_{ s }) coordinate is left blank.

From Fig. 17, we conclude that in 1980 the trapping location of both fine and coarse silt is found at the upstream limit of salt intrusion (i.e., the classical location). In 1980, the sediment is found more into the estuary only for very small river outflow (at most 30 m^{3} s^{ − 1}). Furthermore, we observe two trapping locations only for coarse silt (with settling velocity between 0.001 and 0.002 m s^{ − 1}) and river discharge corresponding to low flow conditions (at most 22 m^{3} s^{ − 1}). In 2005, fine silt is either trapped at the classical location (for very high river discharge of 100–140 m^{3} s^{ − 1}) or in the freshwater zone (river discharge of approximately 30–80 m^{3} s^{ − 1}), which depends on specific magnitude of river discharge *Q* and settling velocity *w* _{ s }. For coarser silt in 2005, two trapping regions are observed even for relatively high river outflow (40–75 m^{3} s^{ − 1}). One ETM is found at the classical location; the other one is located more upstream in the freshwater zone.

## 6 Conclusions

This paper presents a width-averaged analytical model of water and sediment dynamics in an estuary constrained upstream by a weir. Using the morphodynamic equilibrium condition, the trapping of sediment can be described in this estuary. The estuary is assumed to be exponentially converging and vertically well-mixed. The water motion is modeled by the width-averaged shallow water equations, the concentration by the width-averaged advection-diffusion equation. The sediment concentration still depends on a spatially varying erosion coefficient which models the availability of erodible sediment and is found using the condition of morphodynamic equilibrium. Analytic solutions are obtained using a perturbation approach in which physical variables are expanded in power series of the ratio of the semi-diurnal tidal amplitude and the undisturbed water depth at the entrance.

To test the applicability of our model, we consider the Ems estuary. Here, anthropogenic changes altered the water motion and sediment trapping locations significantly between 1980 and 2005. Two distinct bathymetries are considered which represents these two years. Most parameter values are obtained from measurements directly. The vertical eddy viscosity coefficient and stress parameter are obtained by calibrating the model to measured data by minimizing the difference between the observed and modeled semi-diurnal tidal amplitude and the phase difference between the semi-diurnal horizontal and vertical tide in a least square sense. Other observations are used to validate the model. Model results suggest a 50% and 37% reduction in the stress parameter and vertical eddy viscosity coefficient, respectively, in 2005 compared to 1980. The decreased friction and mixing suggests less hydraulic roughness, possibly because bed forms have been removed by dredging or because of the fluid mud that currently covers much of the turbid zone (Talke et al. 2009b).

The model results suggest that the water motion of the estuary is more amplified in 2005. This occurs both due to the deepening of the estuary and to the decreased vertical eddy viscosity and stress parameter. The shift toward greater flood dominance in 2005 is a result of changes in the internally generated overtides. The increased depth and decreased vertical mixing and stress parameter have intensified the gravitational circulation and weakened the tidal return flow in 2005 compared to 1980.

The modeled concentration profiles show that in 1980 both fine and coarse silt is trapped near the upward limit of salt intrusion, with the ETM moving further upstream only for small river outflow (less than 35 m^{3}/s). In 2005, fine silt is either trapped near the upward limit of salt intrusion or in the freshwater zone, depending on the specific magnitude of river discharge and settling velocity. The main import of fine sediment into the estuary is due to the *M* _{2} transport and the major export is caused by the residual transport in both 1980 and 2005. For coarser silt, two trapping regions are observed even for relatively high river outflow (40–75 m^{3} s^{ − 1}). One is found at the classical location, the other one upstream in the freshwater zone. In 1980, the main import of coarse sediment to the ETM is due to the *M* _{2} transport and the export is due to a combination of the residual, *M* _{2} and *M* _{4} transports. In 2005, coarse sediment was primarily imported to the upstream ETM by the *M* _{2} transport and exported by the residual transport, whereas sediment is imported to the downstream ETM due to a combined transport of *M* _{2} and *M* _{4} and exported due to the residual and *M* _{2} transport.

The upstream shift of the estuarine turbidity maximum between 1980 and 2005 and the trapping of sediment in the freshwater zone is primarily a result of changes in tidal asymmetry (the external overtide). The amplification and change of behavior of the externally forced *M* _{4} tide results in a shift of the estuarine turbidity maximum into the freshwater zone in 2005. The difference between the sediment distribution of different grain sizes under the same hydrodynamic conditions (i.e., for the same year) is a result of the temporal settling lag mechanism.

A number of processes that might be important for the trapping of sediment are not taken into account: spatial settling lag, the high sediment concentration, and the SIPS mechanism (strain-induced periodic stratification) discussed for example by (Burchard and Baumert 1998). Note that this kind of model does not allow us to explain an increase of the SSC from 400 to 950 mg/l observed between 1980 and 2005. Furthermore, in our model, the SSC is not found close to the weir. Preliminary results show that the spatial settling lag effect, for example, is important in a region of approximately 20 km downstream from the weir and is negligible in the rest of the estuary. Hence, it probably will enhance the shift of the ETM that is located in the freshwater zone upstream, whereas the ETMs that occur at the classical location will not be affected by this effect.

## Notes

### Acknowledgements

The questions addressed in this manuscript grew out of research funded by LOICZ project 014.27.013 (Land Ocean Interaction in the Coastal Zone), which was administered by NWO-ALW, the Netherlands Organization for Scientific Research. We thank Dr. Martin Krebs and Dr. Helge Juergens from WSA-Emden and Andreas Engels from NLWKN for the data used in the manuscript. Many thanks also to Huib de Swart and Victor de Jonge for their help, comments, and advices.

### **Open Access**

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

## Supplementary material

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