1 

Analysis of tidal straining as driver for estuarine circulation in wellmixed estuaries
Tidal straining, which can mathematically be described as the covariance between eddy viscosity and vertical shear of the alongchannel velocity component, has been acknowledged as one of the major drivers for estuarine circulation in channelized tidally energetic estuaries. In this paper, the authors investigate the role of lateral circulation for generating this covariance. Five numerical experiments are carried out, starting with a reference scenario including the full physics and four scenarios in which specific key physical processes are neglected. These processes are longitudinal internal pressure gradient forcing, lateral internal pressure gradient forcing, lateral advection, and the neglect of temporal variation of eddy viscosity. The results for the viscosity–shear covariance are correlated across different experiments to quantify the change due to neglect of these key processes. It is found that the lateral advection of vertical shear of the alongchannel velocity component and its interaction with the tidally asymmetric eddy viscosity (which is also modified by the lateral circulation) is the major driving force for estuarine circulation in wellmixed tidal estuaries.

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2 

Horizontally viscous effects in a tidal basin: extending Taylor’s problem
The classical problem of Taylor (Proc. Lond. Math. Soc., vol. 20, 1921, pp. 148–181) of Kelvin wave reflection in a semienclosed rectangular basin of uniform depth is extended to account for horizontally viscous effects. To this end, we add horizontally viscous terms to the hydrodynamic model (linearized depthaveraged shallowwater equations on a rotating plane, including bottom friction) and introduce a noslip condition at the closed boundaries.
In a straight channel of infinite length, we obtain three types of wave solutions (normal modes). The first two wave types are viscous Kelvin and Poincaré modes. Compared to their inviscid counterparts, they display longitudinal boundary layers and a slight decrease in the characteristic length scales (wavelength or alongchannel decay distance). For each viscous Poincaré mode, we additionally find a new mode with a nearly similar lateral structure. This third type, entirely due to viscous effects,
represents evanescent waves with an alongchannel decay distance bounded by the boundarylayer thickness.
The solution to the viscous Taylor problem is then written as a superposition of these normal modes: an incoming Kelvin wave and a truncated sum of reflected modes. To satisfy no slip at the lateral boundary, we apply a Galerkin method. The solution displays boundary layers, the lateral one at the basin’s closed end being created by the (new) modes of the third type. Amphidromic points, in the inviscid and frictionless case located on the centreline of the basin, are now found on a line making
a small angle to the longitudinal direction. Using parameter values representative for the Southern Bight of the North Sea, we finally compare the modelled and observed tide propagation in this basin.

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3 

Channel and shoal development in a short tidal embayment: an idealized model study
In many tidal embayments, complex patterns of channels and shoals are observed. To gain a better understanding of these features, an idealized model, that describes the interaction of water motion, sediment transport and bed evolution in a semienclosed, rectangular basin, is developed and analysed. To explain the initial formation of channels and shoals, twodimensional perturbations superposed on a laterally uniform equilibrium bottom are studied. These perturbations evolve due to convergences of various residual suspended sediment fluxes: a diffusive flux, a flux related to the
bed topography, an advective flux resulting from internally generated overtides and an advective flux due to externally prescribed overtides. For most combinations of these fluxes, perturbations start to grow if the bottom friction is strong enough. Their growth is mainly a result of convergences of diffusive and topographically induced sediment fluxes. Advective contributions due to internally generated overtides enhance this growth. If only diffusive sediment fluxes are considered, the underlying equilibrium is always unstable. This can be traced back to the depth dependence of the deposition parameter. Contrary to the results of previous idealized models, the channels and shoals always initiate in the shallow, landward areas. This is explained by the enhanced generation (compared to that in previous models) of frictional torques in shallow regions. The resulting initial channel–shoal formation compares well with results found in complex numerical model studies. The instability mechanism and the location of the initial formation of bottom patterns do not change qualitatively when varying parameters. Changes are mainly related to differences in the underlying
equilibrium profile due to parameter variations.

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4 

Modeling equilibrium bed profiles of short tidal embayments: on the effect of the vertical distribution of suspended sediment and the influence of the boundary conditions
In many tidal embayments, bottom patterns, such as the channelshoal systems of the Wadden Sea,
are observed. To gain understanding of the mechanisms that result in these bottom patterns, an idealized model is developed and analyzed for short tidal embayments.
In this model, the water motion is described by the depth and widthaveraged shallow water equations and forced by a prescribed sea surface elevation at the entrance of the embayment. The bed evolves due to the divergence and convergence of suspended sediment fluxes. To model this suspendedload sediment transport, the threedimensional advection–diffusion equation is integrated over depth and averaged over the width. One of the sediment fluxes in the resulting onedimensional advection–diffusion equation is proportional to the gradient of the local water depth.
In most models, this topographically induced flux is not present. Using standard continuation techniques, morphodynamic equilibria are obtained for different parameter values and forcing conditions. The bathymetry of the resulting equilibrium bed profiles and their dependency on parameters, such as the phase difference between the externally prescribed M2 and M4 tide and the sediment fall velocity, are explained physically With this model, it is then shown that for embayments that are dominated by a net import of sediment, morphodynamic equilibria only exist up to a maximum embayment length. Furthermore, the sensitivity of the model to different morphological boundary conditions at the entrance of the embayment is investigated and it is demonstrated how this strongly influences the shape and number of possible equilibrium bottom profiles.
This paper ends with a comparison between the developed model and field data for the Wadden Sea’s Ameland and Frisian inlets. When the model is forced with the observed M2 and M4 tidal constituents, morphodynamic equilibria can be found with embayment lengths similar to those observed in these inlets. However, this is only possible when the topographically induced suspended sediment flux is included. Without this flux, the maximum embayment length for which morphodynamic equilibria can be found is approximately a third of the observed length. The sensitivity of the model to the topographically induced sediment flux is discussed in detail.

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5 

Influence of topography on tide propagation and amplification in semienclosed basins
An idealized model for tide propagation and amplification in semienclosed rectangular basins is
presented, accounting for depth differences by a combination of longitudinal and lateral topographic steps. The basin geometry is formed by several adjacent compartments of identical width, each having either a uniform depth or two depths separated by a transverse topographic step. The problem is forced by an incoming Kelvin wave at the open end, while allowing waves to radiate outward. The solution in each compartment is written as the superposition of (semi)analytical wave solutions in an infinite channel, individually satisfying the depthaveraged linear shallow water equations on the f plane, including bottom friction. A collocation technique is employed to satisfy continuity of elevation
and flux across the longitudinal topographic steps between the compartments. The model results show that the tidal wave in shallow parts displays slower propagation, enhanced dissipation and amplified amplitudes.
This reveals a resonance mechanism, occurring when the length of the shallow end is roughly an odd multiple of the quarter Kelvin wavelength. Alternatively, for sufficiently wide basins, also Poincaré waves may become resonant. A transverse step implies different wavelengths of the incoming and reflected Kelvin wave, leading to increased amplitudes in shallow regions and a shift of amphidromic points in the direction of the deeper part. Including the shallow parts near the basin’s closed end (thus capturing the Kelvin resonance mechanism) is essential to reproduce semidiurnal and diurnal
tide observations in the Gulf of California, the Adriatic Sea and the Persian Gulf.

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6 

Importance of crosschannel bathymetry and eddy viscosity parameterisation in modelling estuarine flow
For a proper understanding of flow patterns in curved tidal channels, quantification of contributions from
individual physical mechanisms is essential. We study quantitatively how such contributions are affected by crosschannel bathymetry and three alternative eddy viscosity parameterisations. Two models are presented for this purpose, both describing flow in curved but otherwise prismatic channels with an (almost) arbitrary transverse bathymetry. One is a numerical model based on the full threedimensional
shallow water equations. Special feature of this diagnostic model is that assumptions regarding the relative importance of particular physical mechanisms can be incorporated in the computations by switching corresponding terms in the model equations on or off. We also present an idealized model that provides semianalytical approximate solutions of the shallow water equations for all three considered
alternative eddy viscosity parameterisations. It forms an aid in explaining and theorising about results
obtained with the numerical model. Observations regarding Chesapeake Bay serve as a reference case for the present study. We find that the relative importance of both along channel advective forcing and transverse diffusive forcing depends on local characteristics of the crosssectional bottom profile rather than global ones. In our reference case, tideresidual alongchannel flow induced by these forcings
is not small compared to the total tidal residual. Building on this observation, we present an indicative test to judge whether advective processes should be included in leading order in modelling tidedominated estuarine flow. Furthermore, depending on the applied eddy viscosity parameterisation
(uniformly or parabolically distributed over the vertical), we find qualitatively different spatial patterns for
the alongchannel advective forcing.

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7 

Morphodynamic instabilities of planar beaches: Sensitivity to parameter values and process formulations
The initial growth of bed perturbations on planar sloping beaches under the forcing of obliquely incident, breaking waves is investigated using a state‐of‐the‐art numerical model. This allows for a systematic investigation of the sensitivity of the spatial structures of the bed perturbations and their growth and migration rates to different model formulations and parameterizations. If the sediment is only transported in the direction of the net current velocity and sediment stirring is taken proportional to the wave height squared, growing up‐current oriented crescentic bars are found with a preferred spacing of 800 m and a down‐current migration rate of 10 m d−1. Varying the angle of wave incidence, drag coefficient and bed slope results in qualitatively similar growing bed forms. Using an Engelund and Hansen transport formula, very oblique down‐current oriented bars are obtained that grow in time. No preferred wavelength, however, is found. Using the Bailard transport formula results in growing, up‐current oriented bars with a preferred spacing smaller than 300 m for wave angles smaller than 7°. When using either the Engelund and Hansen or Bailard sediment transport formulation, it is essential to take the transport in the direction of the wave orbital velocity into account in order to have growing bed perturbations.

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8 

Morphodynamic evolution of doublebarred beaches
The linear and nonlinear morphological behavior of doublebarred coastal systems under the forcing of obliquely incident waves is studied using a nonlinear numerical model. The linearly most unstable bed forms consist of crescentic patterns (rip channels), whose spacing depends on the magnitude of the longshore current velocity. Using the nonlinear model, six morphodynamic experiments have been performed with various initial bed perturbations in order to assess, among others, the influence of the initial bed perturbation on the morphodynamic evolution. The nonlinear experiments have been pursued well into the nonlinear regime, showing that after a phase of initial exponential growth, a highly dynamic behavior is observed and no equilibrium state is reached. The spacings predicted with the linear stability analysis are observed during the exponential phase of the nonlinear experiments. In the dynamic phase, however, four to seven modes significantly contribute to the resulting bed features. In this final stage, the apparent wavelength of 1000 m of the resulting bed forms on the inner bar is quite insensitive to the initial bed perturbation. On the outer bar it seems that the longer the wavelength of the initial bed perturbation, the longer the wavelength of the final bed forms in the dynamic phase and the larger the migration celerity. In general, the bed forms can be characterized as crescentic or undulating bed patterns. Good correspondence between simulated and observed spacings, shapes and migration celerities are found.

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9 

Impact of estuarine convergence on residual circulation in tidally energetic estuaries and inlets
Estuarine convergence (landward reduction of width and/or depth) is known to have the potential to significantly enhance estuarine circulation, a result theoretically derived under the assumption of constant eddy viscosity. Recent studies of longitudinally uniform energetic tidal channels indicate that tidal straining, a process driven by tidally varying eddy viscosity, is a major driver of estuarine circulation. The combined effect of estuarine convergence and tidal straining is investigated, for the first time, in this paper. The present idealized numerical study shows that estuarine convergence is reducing or even reversing tidal straining circulation in such a way that estuarine circulation can be weakened. This is a counterintuitive hydrodynamic effect of estuarine convergence, which may reduce (rather than increase) upestuary particulate matter transport in estuaries and tidal inlets.

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10 

Subsurface Characterization Using a Cellular Automaton Approach
In this paper, a random Cellular Automaton model is developed to characterise heterogeneity of geological formations. The CAmodel is multilateral and can be easily applied in both two and three dimensions. We demonstrate that conditioning on well data is possible and the method is numerically efficient. To construct the model, the subsurface is subdivided into N cells, with an initial lithology assigned to each cell. Rules to update the current cell states are chosen from a set of rules, independently for each cell. The model converges typically in less than 50 iterations to a steady state or periodic solution. Within one period the realisations exhibit similar statistical properties. The final fraction of the various lithologies can be tuned by choosing the proper initial fractions. In this way, geological knowledge of those fractions can be satisfied.

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11 

Effect of bottom stress formulation on modelled flow and turbidity maxima in crosssections of tidedominated estuaries
A threedimensional numerical model with a prognostic salinity field is used to investigate the effect
of a partial slip bottom boundary condition on lateral flow and sediment distribution in a transect of a tidally dominated channel. The transect has a symmetrical Gaussian crosschannel bottom profile. For a deep, wellmixed, tidally dominated channel, partial slip decreases the relative importance of Coriolis deflection on the generation of crosschannel flow patterns. This has profound implications for the lateral distribution of residual salinity that drives the crosschannel residual circulation pattern. Transverse sediment transport, however, is always found to be governed by a balance between advection of residual sediment concentration by residual lateral flow on the one hand and crosschannel diffusion on the other hand. Hence, the changes in the crosschannel distribution of residual
salinity modify the lateral sediment distribution. For no slip, a single turbidity maximum occurs. In contrast, partial slip gives a gradual transition to a symmetrical density distribution with a turbidity maximum near each bank. For a more shallow, partially mixed tidal channel that represents the James River, a single turbidity maximum at the left bank is found irrespective of the nearbed slip condition. In this case, semidiurnal contributions to sediment distribution and lateral flow play an important role in crosschannel sediment transport. As vertical viscosity and diffusivity are increased, a second maximum at the right bank again exists for partial slip.

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12 

The effect of tidal asymmetry and temporal settling lag on sediment trapping in tidal estuaries
Over decades and centuries, the mean depth of estuaries changes due to sealevel 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 widthaveraged 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 M2 transport and the residual transport for fine sediment (ws = 0.2 mm s−1) and the residual, M2 and M4 transports for coarser sediment (ws = 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.

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13 

The influence of basin geometry on equilibrium and stability of double inlet systems
This study investigates the influence of basin geometry on the crosssectional stability of double inlet systems. The inlet is in equilibrium when the amplitude of the inlet velocities equals the equilibrium velocity (~1 m s1). This equilibrium is stable when after a perturbation the crosssections of both inlets return to their original equilibrium value. The necessary amplitudes of the inlet velocities are obtained using an idealized 2DH hydrodynamic that calculates tidal elevation and flow in a geometry consisting of several adjacent rectangular compartments.
Model results suggest that regardless of the inclusion or exclusion of bottom friction in the basin, stable equilibrium states exist. Qualitatively, the influence of basin geometry does not change the presence of stable equilibrium. Quantitatively, however, taking a basin surface area of 1200 km2, equilibrium values can differ up to a factor 2 depending on the geometry of the basin.

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14 

Drivers of residual estuarine circulation in tidally energetic estuaries: straight and irrotational channels with parabolic cross section
The generation of residual circulation in a tidally energetic estuary with constant longitudinal salinity gradient and parabolic cross section is examined by means of a twodimensional crosssectional numerical model, neglecting river runoff and Stokes drift. It is shown how the longitudinal and lateral residual circulation can be decomposed into contributions from various processes such as tidal straining circulation, gravitational circulation, advectively driven circulation, and horizontal mixing circulation. The sensitivity of the residual circulation and its components from various processes to changes in forcing is investigated by varying the Simpson number (nondimensional longitudinal buoyancy gradient) and the unsteadiness parameter (nondimensional tidal frequency), as well as the bed roughness and the width of the estuary. For relatively weak salinity gradient forcing, the tidal straining circulation dominates the residual exchange circulation in support of classical estuarine circulation (upestuary flow near the bed and downestuary flow near the surface). The strength of the longitudinal estuarine circulation clearly increases with increased salinity gradient forcing. However, when the Simpson number exceeds 0.15, the relative contributions of both gravitational circulation and advectively driven circulation to estuarine circulation increase substantially.
Lateral residual circulation is relatively weak for small Simpson numbers and becomes flood oriented (divergent flow near the bed and convergent flow near the surface) for larger Simpson numbers because of increasing contributions from gravitational and advectively driven circulation. Increasing the unsteadiness number leads to decreased longitudinal and lateral residual circulation. Although changes in bed roughness result in relatively small changes in residual circulation, results are sensitive to the width of the estuary, mainly because of changes in residual exchange circulation driven by tidal straining.

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15 

Observations of barrier island length explained using an exploratory morphodynamic model
Barrier coasts display a chain of islands, separated by tidal inlets that connect a backbarrier basin to a sea or ocean. Observations show that barrier island length generally decreases for increasing tidal range and increasing basin area. However, this has neither been reproduced in model studies nor explained from the underlying physics. This is the aim of our study. Here we simulate barrier coast dynamics by combining a widely used empirical relationship for inlet dynamics with a processbased model of the tidal hydrodynamics. Our model results show stable inlet systems with more than one inlet open that support the observed qualitative relationships and fit in existing barrier coast classifications. To explain this, we identify a competition between a destabilizing mechanism (bottom friction in inlets, tending to reduce the number of open inlets) and a stabilizing one (spatially varying pressure gradients over the inlets, tending to keep the inlets open).

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16 

Residual sediment fluxes in weaklytoperiodically stratified estuaries and tidal inlets
In this idealized numerical modeling study, the composition of residual sediment fluxes in energetic (e.g., weakly or periodically stratified) tidal estuaries is investigated by means of onedimensional water column models, with some focus on the sediment availability. Scaling of the underlying dynamic equations shows dependence of the results on the Simpson number (relative strength of horizontal density gradient) and the Rouse number (relative settling velocity) as well as impacts of the Unsteadiness number (relative tidal frequency). Here, the parameter space given by the Simpson and Rouse numbers is mainly investigated. A simple analytical model based on the assumption of stationarity shows that for small Simpson and Rouse numbers sediment flux is down estuary and vice versa for large Simpson and Rouse numbers. A fully dynamic water column model coupled to a secondmoment turbulence closure model allows to decompose the sediment flux profiles into contributions from the transport flux (product of subtidal velocity and sediment concentration profiles) and the fluctuation flux profiles (tidal covariance between current velocity and sediment concentration). Three different types of bottom sediment pools are distinguished to vary the sediment availability, by defining a time scale for complete sediment erosion. For short erosion times scales, the transport sediment flux may dominate, but for larger erosion time scales the fluctuation sediment flux largely dominates the tidal sediment flux. When quarterdiurnal components are added to the tidal forcing, upestuary sediment fluxes are strongly increased for stronger and shorter flood tides and vice versa. The theoretical results are compared to field observations in a tidally energetic inlet.

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17 

Resonance characteristics of tides in branching channels

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18 

Seasonal variability in M2 and M4 tidal constituents and its implications for the coastal residual sediment transport
We use an observational data set of tidal gauges in the North Sea to investigate the annual cycle of the M2 and M4 amplitudes and phases. The sea surface elevation amplitude of the M2 can vary by 8–10% and the M4 amplitude by 12–30% over the course of the year, with larger amplitudes in summer. The annual phase variations are in the range of 3–15◦. The reason for these variations is the thermal structure of the North Sea: a welldeveloped thermocline in summer and wellmixed water column during winter. The interaction of the M2 and M4 tides is one of the main drivers of the residual sediment transport. Using an analytical model, the seasonal variability in residual sediment transport is estimated. This transport can vary by 10–50% over the course of the year. These variations are mainly related to the seasonal variability of the M2 and M4 amplitudes.

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19 

Instability of timedependent winddriven ocean gyres
The winddriven ocean circulation at midlatitudes is susceptible to several types of instabilities. One of the simplest models of these flows is the quasigeostrophic barotropic potential vorticity equation in an idealized ocean basin. In this model, the route to complex spatio/temporal flows is through successive bifurcations. The aim of this study is to describe the physics of the destabilization process of a periodic winddriven flow associated with a secondary bifurcation. Although bifurcation theory has proven to be a valuable tool to determine the physical mechanisms of destabilization of fluid flows, the analysis of the stability of timedependent (for example, periodic) flows, using this methodology, is computationally unpractical, due to the large number of degreesoffreedom involved. The approach followed here is to construct a loworder model using numerical Galerkin projection of the full model equations onto the dynamically active eigenmodes. The resulting reduced model is shown to capture the local dynamics of the full model. The physical mechanism of the destabilization of the periodic winddriven flow is deduced from the reduced model. While there are several stabilizing processes, notably rectification, the destabilization occurs due to timedependent increase of the background horizontal shear in the flow.

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20 

Lateral entrapment of sediment in tidal estuaries: An idealized model study
Two physical mechanisms leading to lateral accumulation of sediment in tidally dominated estuaries are investigated, involving Coriolis forcing and lateral density gradients. An idealized model is used that consists of the three‐dimensional shallow water equations and sediment mass balance. Conditions are assumed to be uniform in the along‐estuary direction. A semidiurnal tidal discharge and tidally averaged density gradients are prescribed. The erosional sediment flux at the bed depends both on the bed shear stress and on the amount of sediment available in mud reaches for resuspension. The distribution of mud reaches over the bed is selected such that sediment transport is in morphodynamic equilibrium, that is, tidally averaged erosion and deposition of sediment at the bed balance. Analytical solutions are obtained by using perturbation analysis. Results suggest that in most estuaries lateral density gradients induce more sediment transport than Coriolis forcing. When frictional forces are small (Ekman number E < 0.02), the Coriolis mechanism dominates and accumulates sediment on the right bank (looking up‐estuary in the Northern Hemisphere). On the other hand, when frictional forces are moderate to high (E > 0.02), the lateral density gradient mechanism dominates and entraps sediment in areas with fresher water. Results also show that the lateral sediment transport induced by the semidiurnal tidal flow is significant when frictional forces are small (E ∼ 0.02). Model predictions are in good agreement with observations from the James River estuary.

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