Barrier coast systems connect oceanic and inland waters and support essential ecological functions. Their shallow subtidal and intertidal zones provide key ecosystem services, making it crucial to understand the nonlinear mechanisms driving water motion in these regions, particularly in terms of overtides, mean sea surface elevation, and residual flow. This study introduces a new interpretation framework for analyzing nonlinear processes in tidal systems with shallow regions, extending the established framework by Parker (1991) to account for wetting and drying dynamics. To illustrate this framework, it is applied to a model of the Marsdiep-Vlie double-inlet system in the Wadden Sea, governed by the cross-sectionally averaged shallow water equations, including the Defina (2000) approach to account for wetting and drying. The one-dimensional model bathymetry and planform feature a topographical high and varying channel widths, respectively. We assess contributions of nonlinear forcings to overtides and residual flow, and conduct sensitivity analyses on the effects of longitudinal variations in bottom elevation and channel width. Results confirm that Parker’s findings hold in deeper systems with minimal width variation; however, in systems with intertidal areas, distinct nonlinear processes emerge. Specifically, in these shallow regions, the Water Storage term, Nonlinear Continuity term, and Elevation Effect of the Friction term contribute greatly to generation, while the Nonlinear Continuity term strongly contributes to generation. For mean sea surface elevation, several nonlinear terms are equally important, while residual velocities are predominantly driven by the Nonlinear Continuity term.

Nonlinear tidal processes in water motion, which generate overtides and residual dynamics, are crucial not only for the morphodynamic evolution of tidal systems, but also for the transport of contaminants, nutrients and biogeochemical tracers. Traditional approaches, such as that of Parker (1991), explain these nonlinear processes using perturbation methods that assume a small tidal amplitude-to-depth ratio. While these methods effectively detect nonlinear mechanisms, quantifying their relative importance in practical applications remains challenging. Moreover, this assumption limits their applicability in very shallow and intertidal areas, where it is violated. To address this limitation, we propose a new methodology to assess the relative importance of overtide generation mechanisms. This approach is based on the fully nonlinear shallow water equations, combined with Defina’s (2000) wetting and drying approach to account for subgrid-scale topography. As a first validation step of this methodology, we applied it to a schematization of the Delaware estuary, the same case studied by Parker (1991), where very shallow or intertidal areas are absent. This setup allows for a direct comparison, particularly for the cases where an M₂ tidal constituent was imposed at the seaward boundary. Our analysis confirms the dominant role of the nonlinear continuity term in the M₄ overtide generation, while the combined velocity and sea surface elevation nonlinearity in bed friction plays a secondary role. For the M6 overtide, the quadratic velocity friction term, which accounts only for velocity nonlinearity, plays a pivotal role. Additionally, the combined velocity and sea surface elevation nonlinearity in bed friction is the main driver of mean sea level generation. The inclusion of mean Delaware River runoff enhances the M4 generation while slightly reducing the amplitudes of the M6 tidal constituent, and it leads to an increase in mean sea surface elevation. The observations above are consistent with Parker's results. In future research, the new methodology will be further explored for tidal systems with extensive intertidal areas, where Parker’s approach is no longer applicable.

This study examines the local, intratidal effects of suspended sediment concentrations (SSCs) on the hydrodynamics and vertical mixing in the Ems Estuary, located on the border between Germany and The Netherlands, during summer and winter seasons when the estuary turbidity maximum (ETM) is located upstream and adjacent to the study site, respectively. Measurements of density, SSCs, turbulent kinetic energy dissipation, and current velocity were collected and analyzed over a semi-diurnal tidal cycle in August of 2018 and January of 2019 as part of the collaborative Ems-Dollard Measurement (EDoM) campaign. During August, the estuary turbidity maximum was located 25 km upstream from the measurement site and local SSCs were low. Results revealed that under these conditions, suspended sediment minimally impacted vertical mixing by stabilizing density near-bottom during flood tide, while typical salinity-induced tidal straining patterns dominated. During January, the ETM was located only 5 km upstream of the measurement site leading to higher local sediment concentrations. Salinity-induced straining of the density occurred on early flood tide, creating stratification that suppressed vertical mixing. The suppression was enhanced by the contribution of vertical gradients in SSC to density, as signified by the gradient Richardson number. Suppression of vertical mixing by sediment-enhanced stratification was most significant within the hour following maximum flood currents when elevated velocity shear occurred. The variability observed between the local dynamics during August and January were attributed to greater sediment concentrations due to the ETM proximity in January. The intratidal asymmetry of vertical mixing observed under higher SSCs likely has implications for sediment transport.

The water motion computed using 3D and 2DH models in tidally dominated shallow waters can, in some cases, differ significantly. In 2DH models, bed friction is typically parametrised in terms of the depth-averaged velocity, whereas in 3D models, typically the near-bed velocity is used. This difference causes the bed shear stress in 2DH models to point towards the depth-averaged velocity, whereas in 3D models, it points towards the near-bed velocity, which are not necessarily the same. Focussing on linearised barotropic models, we derive an exact friction parametrisation for 2DH models such that the same depth-averaged dynamics are described as in the corresponding 3D model. The result is a convolutional friction formulation where the instantaneous friction depends on the present and past velocities, thus modifying the traditional 2DH friction formulation that only depends on the present depth-averaged velocity. In the case of harmonic (tidal) waves, this parametrisation has a clear physical interpretation and shows that the near-bed velocity should be parametrised as a rotated, deformed and phase shifted variant of the depth-averaged velocity. We demonstrate that in certain regions of the parameter space, it may be impossible to calibrate a 2DH model that uses a traditional friction law to reproduce the water levels from a 3D model, showing that the 3D friction formulation can be crucial to capture the 3D dynamics within a depth-averaged model. This phenomenon is explored in detail in a narrow well-mixed estuary.

Structure and intensity of estuarine exchange flow depend significantly on the eddy viscosity A_v profile which is dynamically linked to various forces (e.g., gravitational, tidal, wind-driven). The impact of winds on the exchange flow is complex due to its direct (local and remote changes in shear and density stratification) and indirect (modifications to A_v profiles) contributions. This study aims (i) to include wind entrainment effects in the tidally averaged A_v parameterization; (ii) to develop an analytical one-dimensional model for the wind driven exchange flow by using this novel parameterization and assess the tidally averaged dynamics over a relevant physical parameter-space, subdomains of which have not yet been explored numerically. This one-dimensional model is based on a balance between frictional forces and pressure gradient, calibrated with a tidally-resolving one-dimensional water-column model with second-moment closure. Structure and intensity of the resulting exchange flow profiles are analyzed with respect to three dimensionless parameters (the unsteadiness of boundary layer mixing U_n, scaled-directional wind stress W, and horizontal stratification S_i). While down-estuarine winds enhance the gravitational circulation, up-estuarine winds result in either a two-layer inverted circulation opposing the gravitational circulation, or a three-layer flow (favored by relatively strong S_i, weak W, and moderate U_n) that is up-estuarine at the surface with classical two-layer circulation underneath. Relative thickness of surface and bottom boundary layers affect both the intensity and the inflection depth of the exchange flow layers. Up-estuarine winds with W≳0.5 yield unstable stratification and reduce the exchange flow intensity with increasing W.

Tidally averaged transport of salt in estuaries is controlled by various subtidal and tidal processes. In this study, we show the relative importance of various subtidal and tidal transport processes in a width-averaged sense. This is done for a large range of forcing and geometric parameters, which describe well-mixed to salt wedge estuaries. To this end, we develop a width-averaged process-based model aimed at conducting and analyzing a large number of experiments (∼40,000). We find that the salt transport is dominated by one of seven salt transport balances, or regimes. Four of these regimes are dominated by subtidal processes, while the other three are dominated by tidal processes. Which regime occurs in a part of an estuary depends on four dimensionless parameters, representing local geometry, and forcing conditions. One of the regimes features salt import by correlations between the depth-averaged tidal velocity and salinity. While this mechanism was previously only associated with along-channel geometric variations, we find it can also be a dominant mechanism in a significant part of the parameter space due to river-induced tidal asymmetry, independent of river geometry. We apply our classification to a case study of part of the Dutch Rhine delta and compare to decomposition results of a fully realistic three-dimensional model. We find the estuary features two regimes, with import dominated by subtidal shear transport in the seaward part of the estuary and by depth-averaged tidal correlations in the landward part of the estuary.

An idealized width-averaged model is employed to study the influence of wind stress on subtidal salt intrusion and stratification in well-mixed and partially stratified estuaries. We show that even in mild conditions, wind forcing can influence the estuarine salinity structure in a substantial way. By studying the role of wind forcing on dominant salt transport balances and associated salt transport regimes, we unify and clarify ambiguous observations from previous authors regarding the influence of wind stress: the response of the estuarine salinity structure to wind forcing is different depending on the underlying dominant salt transport balance, which in turn was found to determine whether wind-induced salinity shear, wind-induced modulation of the longitudinal salt distribution, or wind-induced mixing dominates. SIGNIFICANCE STATEMENT: The purpose of this idealized study is to better understand how wind influences the salinity distribution in estuaries on large time scales. This is important because a change in winds can move saline water further inland, threatening freshwater availability and the natural balance of delicate ecosystems. We clarify the sometimes ambiguous observations regarding the influence of wind on the salt distribution and highlight the importance of including average wind forcing in analyses of estuarine dynamics on large time scales.

Channel–shoal patterns are often observed in the back–barrier basins of inlet systems and are important from both an economical and ecological point of view. Focussing on double–inlet systems, the initial formation of these patterns is investigated using an idealized model. The model is governed by the depth–averaged shallow water equations, a depth–integrated concentration equation and a tidally–averaged bottom evolution equation. Focussing on rectangular basins and neglecting the effects of earth rotation, it is found that laterally uniform morphodynamic equilibria can become linearly unstable, resulting in initial patterns that resemble channels and shoals. When the water motion is only forced by an M₂ tidal constituent, the existence of (laterally uniform) morphodynamic equilibria for which both inlets are connected strongly depends on the relative phase and amplitudes of the tidal forcing. If such equilibria exist, they can be either stable against small perturbations or linearly unstable. If these equilibria are linearly unstable, two instability mechanisms can be identified, the first related to the convergences and divergences of diffusive transports, the second mechanism related to a combination of advective and diffusive transports. In the former case, all eigenvalues are real and the bedforms grow exponentially in time. In the latter case, the eigenvalues are complex, resulting in bedforms that both migrate and grow in time. In case external overtides and a time–independent discharge are included, no diffusive instabilities are found anymore for the parameters considered in this paper. This implies that all instabilities are migrating in time. In all cases considered, the bed perturbations have only an appreciable amplitude at locations where the underlying laterally uniform equilibrium has a local minimum in water depth. This is consistent with observations from numerical models and laboratory experiments.

The existence of morphodynamic equilibria of double-inlet systems is investigated using a cross-sectionally averaged morphodynamic model. The number of possible equilibria and their stability strongly depend on the forcing conditions and geometry considered. This is illustrated by considering a rectangular double-inlet system forced by M₂ tidal constituents only. Depending on the M₂ amplitudes and phases at both entrances, no equilibrium, one equilibrium or multiple morphodynamic equilibria may exist. In case no equilibrium is found, the minimum water depth becomes zero somewhere in the system, reducing the double-inlet system to two single-inlet systems. In the other cases, the location of the minimum water depth and the direction of the tidally-averaged sediment transport, as well as their actual values, depend strongly on the M₂ tidal characteristics. Such parameter sensitivity is also observed when including the residual and M₄ forcing contributions to the water motion, and when allowing for width variations. This suggests that, when considering a specific system, the number and stability of morphodynamic equilibria, as well as the characteristics of these quantities, can only be assessed by investigating that specific system in detail. As an example, the Marsdiep-Vlie inlet system in the Dutch Wadden Sea is considered. It is found that, by using parameter values and a geometry characteristic for this system, the water motion and bathymetry in morphodynamic equilibrium are qualitatively reproduced. Also the direction and order of magnitude of the tidally-averaged suspended sediment transport compare well with those obtained from a high-complexity numerical model.

A depth-averaged (2DH) exploratory model is developed to identify morphodynamic equilibria in short mesotidal inlet systems with arbitrary planform geometries. The water motion is forced by an M ₂ tidal constituent at the seaward entrance and is described by the depth-averaged shallow water equations, whereas the depth-integrated suspended sediment concentration follows from a diffusion equation, taking into account local inertia, horizontal eddy diffusion and topographically induced diffusive effects, erosion, and deposition. Based on a scaling analysis, it follows that the fine sandy bed evolution is dominated by the depth-integrated diffusive sediment transport. The depth-integrated advective contributions are one order smaller and therefore neglected. This observation also allows for the neglect of the advective terms in the governing equations. The associated morphodynamic equilibria are directly identified based on a continuation approach. By means of the exploratory model, the morphodynamic equilibria are studied in basins with a planform geometry characterized by width variations as a function of the distance to the seaward boundary. The model results show that in the case of a sufficient degree of widening in the landward direction, the equilibrium bed level exhibits significant lateral structures, characterized by shallow zones and deeper channels. The first channel bifurcation, as observed in many short tidal inlet systems, is forced by the planform geometry of the basin, and the associated physical mechanisms are explained. Furthermore, two mechanisms inducing asymmetric morphodynamic equilibria are investigated, of which the effect of an asymmetric basin planform seems to be dominant over that of the Coriolis force.

The salinity structure in estuaries is classically described in terms of the salinity structure as well mixed, partially mixed, or salt wedge. The existing knowledge about the processes that result in such salinity structures comes from highly idealized models that are restricted to either well-mixed and partially mixed cases or subtidal salt wedge estuaries. Hence, there is still little knowledge about the processes driving transitions between these different salinity structures and the estuarine parameters at which such a transition is found. As an important step toward a unified description of the dominant processes driving well-mixed, partially mixed, and salt wedge estuaries, a subtidal width-averaged model applicable to all these salinity structures is developed and systematically analyzed. Using our model, we identify four salinity regimes, resulting from different balances of dominant processes. It is shown that each regime is uniquely determined by two dimensionless parameters: an estuarine Froude and Rayleigh number, representing freshwater discharge and tidal mixing, respectively, resulting in a classification of the regimes in terms of these two parameters. Furthermore, analytical expressions to approximate the salt intrusion length in each regime are developed. These expressions are used to illustrate that the salt intrusion length in different regimes responds in a highly different manner to changes in depth and freshwater discharge. As one of the key results, we show that there are only very weak relations between the process-based regime of an estuary and the salt intrusion length and top-bottom stratification. This implies that the salinity structure of an estuary cannot be uniquely matched to a regime.

Many estuaries exhibit seasonality in the estuary-scale distribution of suspended particulate matter (SPM). This SPM distribution depends on various factors, including freshwater discharge, salinity intrusion, erodibility, and the ability of cohesive SPM to flocculate into larger aggregates. Various authors indicate that biotic factors, such as the presence of algae and their excretion of sticky transparent exopolymer particles (TEP), affect the flocculation and erosion processes. Consequently, seasonality in these biotic factors may play a role in the observed seasonality in SPM. Whereas the impact of abiotic factors on seasonality in SPM is well studied, the relative contribution of biotically induced seasonality is largely unknown. In this study, we employ two approaches to assess the aggregated importance of biotically induced seasonality in flocculation and erosion on seasonality in SPM in the Scheldt estuary. In the first approach, we focus on seasonality of in situ observations in the Scheldt estuary of turbidity, floc size, Chlorophyll-a, and TEP, showing that the abiotic parameters show seasonality, while seasonality in TEP is ambiguous. The second approach concerns a reverse engineering method to calibrate biotically affected parameters of a coupled sediment transport-flocculation model to turbidity observations, allowing us to compare the modeled SPM concentrations to the observations. Driven by seasonality in freshwater discharge, the model captures the observed seasonality in SPM without requiring biotically induced seasonality in flocculation and erosion, which is supported by the absence of seasonality in TEP.

A new parameterization for the estuarine turbulent eddy viscosity coefficient is developed considering the influence of wind forcing and feedback between stratification and shear. The emerging tidally averaged eddy viscosity profile A_v^T is parameterized as parabolic under well-mixed conditions, and is composed of a skewed-Gaussian-like form for the upper layer, and a parabolic form for the bottom layer under stratified conditions. The precise shape of the profiles depends parametrically on the bottom boundary layer thickness, the bulk Richardson number, and the Wedderburn number. The parameterized A_v^T profiles show excellent agreement with profiles obtained from numerical models. To explore the importance of vertically varying A_v^T with regard to exchange processes, an analytical model is designed. This one-dimensional model is based on a balance between frictional forces and pressure gradient. The resulting exchange flow is analyzed over the relevant parameter space that is associated with horizontal and vertical stratification through the bulk Richardson number, and the bi-directional wind stress via the Wedderburn number. Down-estuary wind enhances the up-estuary flow near the bottom and down-estuary flow near the surface driving an exchange flow pattern typically associated with gravitational circulation. Up-estuary wind results in either a two-layer inverted circulation opposing the gravitational circulation, or a three-layer flow that is up-estuarine at the surface with classical two-layer circulation underneath. Three-layer flow emerges with a weak wind. With increasing runoff velocity, three-layer flow transitions to a single layer flow under weak stratification conditions.

In finite element methods, the accuracy of the solution cannot increase indefinitely since the round-off error related to limited computer precision increases when the number of degrees of freedom (DoFs) is large enough. Because a priori information of the highest attainable accuracy is of great interest, we construct an innovative method to obtain the highest attainable accuracy given the order of the elements. In this method, the truncation error is extrapolated when it converges at the asymptotic rate, and the bound of the round-off error follows from a generically valid error estimate, obtained and validated through extensive numerical experiments. The highest attainable accuracy is obtained by minimizing the sum of these two types of errors. We validate this method using a one-dimensional Helmholtz equation in space. It shows that the highest attainable accuracy can be accurately predicted, and the CPU time required is much smaller compared with that using successive grid refinement.